Theorem canthp1lem2 9475

Description: Lemma for canthp1 9476. (Contributed by Mario Carneiro, 18-May-2015.)

Hypotheses

Ref	Expression
canthp1lem2.1	⊢ (𝜑 → 1𝑜 ≺ 𝐴)
canthp1lem2.2	⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜))
canthp1lem2.3	⊢ (𝜑 → 𝐺:((𝐴 +𝑐 1𝑜) ∖ {(𝐹‘𝐴)})–1-1-onto→𝐴)
canthp1lem2.4	⊢ 𝐻 = ((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))
canthp1lem2.5	⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐻‘(◡𝑟 “ {𝑦})) = 𝑦))}
canthp1lem2.6	⊢ 𝐵 = ∪ dom 𝑊

Assertion

Ref	Expression
canthp1lem2	⊢ ¬ 𝜑

Distinct variable groups:   𝑥,𝑟,𝑦,𝐴   𝐵,𝑟,𝑥,𝑦   𝐻,𝑟,𝑥,𝑦   𝜑,𝑟,𝑥,𝑦   𝑊,𝑟,𝑥,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦,𝑟)   𝐺(𝑥,𝑦,𝑟)

Proof of Theorem canthp1lem2

Step	Hyp	Ref	Expression
1		canthp1lem2.1	. . . . . 6 ⊢ (𝜑 → 1𝑜 ≺ 𝐴)
2		relsdom 7962	. . . . . . 7 ⊢ Rel ≺
3	2	brrelex2i 5159	. . . . . 6 ⊢ (1𝑜 ≺ 𝐴 → 𝐴 ∈ V)
4	1, 3	syl 17	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
5		pwexg 4850	. . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
6	4, 5	syl 17	. . . 4 ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
7		canthp1lem2.2	. . . 4 ⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜))
8		f1oeng 7974	. . . 4 ⊢ ((𝒫 𝐴 ∈ V ∧ 𝐹:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜)) → 𝒫 𝐴 ≈ (𝐴 +𝑐 1𝑜))
9	6, 7, 8	syl2anc 693	. . 3 ⊢ (𝜑 → 𝒫 𝐴 ≈ (𝐴 +𝑐 1𝑜))
10		ensym 8005	. . 3 ⊢ (𝒫 𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) ≈ 𝒫 𝐴)
11	9, 10	syl 17	. 2 ⊢ (𝜑 → (𝐴 +𝑐 1𝑜) ≈ 𝒫 𝐴)
12		canth2g 8114	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
13	4, 12	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≺ 𝒫 𝐴)
14		sdomen2 8105	. . . . . . . . . . 11 ⊢ (𝒫 𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 ≺ 𝒫 𝐴 ↔ 𝐴 ≺ (𝐴 +𝑐 1𝑜)))
15	9, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ≺ 𝒫 𝐴 ↔ 𝐴 ≺ (𝐴 +𝑐 1𝑜)))
16	13, 15	mpbid 222	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≺ (𝐴 +𝑐 1𝑜))
17		sdomnen 7984	. . . . . . . . 9 ⊢ (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
18	16, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
19		omelon 8543	. . . . . . . . . . . 12 ⊢ ω ∈ On
20		onenon 8775	. . . . . . . . . . . 12 ⊢ (ω ∈ On → ω ∈ dom card)
21	19, 20	ax-mp 5	. . . . . . . . . . 11 ⊢ ω ∈ dom card
22		canthp1lem2.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐺:((𝐴 +𝑐 1𝑜) ∖ {(𝐹‘𝐴)})–1-1-onto→𝐴)
23		dff1o3 6143	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜) ↔ (𝐹:𝒫 𝐴–onto→(𝐴 +𝑐 1𝑜) ∧ Fun ◡𝐹))
24	23	simprbi 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜) → Fun ◡𝐹)
25	7, 24	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → Fun ◡𝐹)
26		f1ofo 6144	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜) → 𝐹:𝒫 𝐴–onto→(𝐴 +𝑐 1𝑜))
27	7, 26	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹:𝒫 𝐴–onto→(𝐴 +𝑐 1𝑜))
28		f1ofn 6138	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:𝒫 𝐴–1-1-onto→(𝐴 +𝑐 1𝑜) → 𝐹 Fn 𝒫 𝐴)
29		fnresdm 6000	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 Fn 𝒫 𝐴 → (𝐹 ↾ 𝒫 𝐴) = 𝐹)
30		foeq1 6111	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 ↾ 𝒫 𝐴) = 𝐹 → ((𝐹 ↾ 𝒫 𝐴):𝒫 𝐴–onto→(𝐴 +𝑐 1𝑜) ↔ 𝐹:𝒫 𝐴–onto→(𝐴 +𝑐 1𝑜)))
31	7, 28, 29, 30	4syl 19	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((𝐹 ↾ 𝒫 𝐴):𝒫 𝐴–onto→(𝐴 +𝑐 1𝑜) ↔ 𝐹:𝒫 𝐴–onto→(𝐴 +𝑐 1𝑜)))
32	27, 31	mpbird 247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐹 ↾ 𝒫 𝐴):𝒫 𝐴–onto→(𝐴 +𝑐 1𝑜))
33		fvex 6201	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹‘𝐴) ∈ V
34		f1osng 6177	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ V ∧ (𝐹‘𝐴) ∈ V) → {⟨𝐴, (𝐹‘𝐴)⟩}:{𝐴}–1-1-onto→{(𝐹‘𝐴)})
35	4, 33, 34	sylancl 694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {⟨𝐴, (𝐹‘𝐴)⟩}:{𝐴}–1-1-onto→{(𝐹‘𝐴)})
36	7, 28	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐹 Fn 𝒫 𝐴)
37		pwidg 4173	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
38	4, 37	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐴)
39		fnressn 6425	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹 Fn 𝒫 𝐴 ∧ 𝐴 ∈ 𝒫 𝐴) → (𝐹 ↾ {𝐴}) = {⟨𝐴, (𝐹‘𝐴)⟩})
40	36, 38, 39	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝐹 ↾ {𝐴}) = {⟨𝐴, (𝐹‘𝐴)⟩})
41		f1oeq1 6127	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹 ↾ {𝐴}) = {⟨𝐴, (𝐹‘𝐴)⟩} → ((𝐹 ↾ {𝐴}):{𝐴}–1-1-onto→{(𝐹‘𝐴)} ↔ {⟨𝐴, (𝐹‘𝐴)⟩}:{𝐴}–1-1-onto→{(𝐹‘𝐴)}))
42	40, 41	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝐹 ↾ {𝐴}):{𝐴}–1-1-onto→{(𝐹‘𝐴)} ↔ {⟨𝐴, (𝐹‘𝐴)⟩}:{𝐴}–1-1-onto→{(𝐹‘𝐴)}))
43	35, 42	mpbird 247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐹 ↾ {𝐴}):{𝐴}–1-1-onto→{(𝐹‘𝐴)})
44		f1ofo 6144	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 ↾ {𝐴}):{𝐴}–1-1-onto→{(𝐹‘𝐴)} → (𝐹 ↾ {𝐴}):{𝐴}–onto→{(𝐹‘𝐴)})
45	43, 44	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐹 ↾ {𝐴}):{𝐴}–onto→{(𝐹‘𝐴)})
46		resdif 6157	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝒫 𝐴):𝒫 𝐴–onto→(𝐴 +𝑐 1𝑜) ∧ (𝐹 ↾ {𝐴}):{𝐴}–onto→{(𝐹‘𝐴)}) → (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→((𝐴 +𝑐 1𝑜) ∖ {(𝐹‘𝐴)}))
47	25, 32, 45, 46	syl3anc 1326	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→((𝐴 +𝑐 1𝑜) ∖ {(𝐹‘𝐴)}))
48		f1oco 6159	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺:((𝐴 +𝑐 1𝑜) ∖ {(𝐹‘𝐴)})–1-1-onto→𝐴 ∧ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→((𝐴 +𝑐 1𝑜) ∖ {(𝐹‘𝐴)})) → (𝐺 ∘ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴}))):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→𝐴)
49	22, 47, 48	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺 ∘ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴}))):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→𝐴)
50		resco 5639	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) = (𝐺 ∘ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴})))
51		f1oeq1 6127	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) = (𝐺 ∘ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴}))) → (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→𝐴 ↔ (𝐺 ∘ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴}))):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→𝐴))
52	50, 51	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→𝐴 ↔ (𝐺 ∘ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴}))):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→𝐴)
53	49, 52	sylibr 224	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→𝐴)
54		f1of 6137	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→𝐴 → ((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})⟶𝐴)
55	53, 54	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})⟶𝐴)
56		0elpw 4834	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ ∈ 𝒫 𝐴
57	56	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 = 𝐴) → ∅ ∈ 𝒫 𝐴)
58		sdom0 8092	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ¬ 1𝑜 ≺ ∅
59		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∅ = 𝐴 → (1𝑜 ≺ ∅ ↔ 1𝑜 ≺ 𝐴))
60	58, 59	mtbii 316	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ = 𝐴 → ¬ 1𝑜 ≺ 𝐴)
61	60	necon2ai 2823	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1𝑜 ≺ 𝐴 → ∅ ≠ 𝐴)
62	1, 61	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ∅ ≠ 𝐴)
63	62	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 = 𝐴) → ∅ ≠ 𝐴)
64		eldifsn 4317	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ ∈ (𝒫 𝐴 ∖ {𝐴}) ↔ (∅ ∈ 𝒫 𝐴 ∧ ∅ ≠ 𝐴))
65	57, 63, 64	sylanbrc 698	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 = 𝐴) → ∅ ∈ (𝒫 𝐴 ∖ {𝐴}))
66		simplr 792	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ∈ 𝒫 𝐴)
67		simpr 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ ¬ 𝑥 = 𝐴) → ¬ 𝑥 = 𝐴)
68	67	neqned 2801	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ≠ 𝐴)
69		eldifsn 4317	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝒫 𝐴 ∖ {𝐴}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ≠ 𝐴))
70	66, 68, 69	sylanbrc 698	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ∈ (𝒫 𝐴 ∖ {𝐴}))
71	65, 70	ifclda 4120	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → if(𝑥 = 𝐴, ∅, 𝑥) ∈ (𝒫 𝐴 ∖ {𝐴}))
72		eqid 2622	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)) = (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))
73	71, 72	fmptd 6385	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)):𝒫 𝐴⟶(𝒫 𝐴 ∖ {𝐴}))
74		fco 6058	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})⟶𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)):𝒫 𝐴⟶(𝒫 𝐴 ∖ {𝐴})) → (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))):𝒫 𝐴⟶𝐴)
75	55, 73, 74	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))):𝒫 𝐴⟶𝐴)
76		frn 6053	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)):𝒫 𝐴⟶(𝒫 𝐴 ∖ {𝐴}) → ran (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)) ⊆ (𝒫 𝐴 ∖ {𝐴}))
77	73, 76	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)) ⊆ (𝒫 𝐴 ∖ {𝐴}))
78		cores 5638	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)) ⊆ (𝒫 𝐴 ∖ {𝐴}) → (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))) = ((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))))
79	77, 78	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))) = ((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))))
80		canthp1lem2.4	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐻 = ((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))
81	79, 80	syl6eqr 2674	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))) = 𝐻)
82	81	feq1d 6030	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))):𝒫 𝐴⟶𝐴 ↔ 𝐻:𝒫 𝐴⟶𝐴))
83	75, 82	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:𝒫 𝐴⟶𝐴)
84		inss1 3833	. . . . . . . . . . . . . . . 16 ⊢ (𝒫 𝐴 ∩ dom card) ⊆ 𝒫 𝐴
85	84	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝒫 𝐴 ∩ dom card) ⊆ 𝒫 𝐴)
86		canthp1lem2.5	. . . . . . . . . . . . . . . 16 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐻‘(◡𝑟 “ {𝑦})) = 𝑦))}
87		canthp1lem2.6	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = ∪ dom 𝑊
88		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) = (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})
89	86, 87, 88	canth4 9469	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ V ∧ 𝐻:𝒫 𝐴⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝒫 𝐴) → (𝐵 ⊆ 𝐴 ∧ (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ⊊ 𝐵 ∧ (𝐻‘𝐵) = (𝐻‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}))))
90	4, 83, 85, 89	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 ⊆ 𝐴 ∧ (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ⊊ 𝐵 ∧ (𝐻‘𝐵) = (𝐻‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}))))
91	90	simp1d 1073	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
92	90	simp2d 1074	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ⊊ 𝐵)
93	92	pssned 3705	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ≠ 𝐵)
94	93	necomd 2849	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}))
95	90	simp3d 1075	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐻‘𝐵) = (𝐻‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
96	80	fveq1i 6192	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐻‘𝐵) = (((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘𝐵)
97	80	fveq1i 6192	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐻‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) = (((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}))
98	95, 96, 97	3eqtr3g 2679	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘𝐵) = (((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
99		elpw2g 4827	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
100	4, 99	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
101	91, 100	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐴)
102		fvco3 6275	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)):𝒫 𝐴⟶(𝒫 𝐴 ∖ {𝐴}) ∧ 𝐵 ∈ 𝒫 𝐴) → (((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘𝐵) = ((𝐺 ∘ 𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵)))
103	73, 101, 102	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘𝐵) = ((𝐺 ∘ 𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵)))
104	92	pssssd 3704	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ⊆ 𝐵)
105	104, 91	sstrd 3613	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ⊆ 𝐴)
106		elpw2g 4827	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ V → ((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ∈ 𝒫 𝐴 ↔ (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ⊆ 𝐴))
107	4, 106	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ∈ 𝒫 𝐴 ↔ (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ⊆ 𝐴))
108	105, 107	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ∈ 𝒫 𝐴)
109		fvco3 6275	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)):𝒫 𝐴⟶(𝒫 𝐴 ∖ {𝐴}) ∧ (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ∈ 𝒫 𝐴) → (((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) = ((𝐺 ∘ 𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}))))
110	73, 108, 109	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) = ((𝐺 ∘ 𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}))))
111	98, 103, 110	3eqtr3d 2664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵)) = ((𝐺 ∘ 𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}))))
112	111	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → ((𝐺 ∘ 𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵)) = ((𝐺 ∘ 𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}))))
113		ifcl 4130	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝐵 ∈ 𝒫 𝐴) → if(𝐵 = 𝐴, ∅, 𝐵) ∈ 𝒫 𝐴)
114	56, 101, 113	sylancr 695	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → if(𝐵 = 𝐴, ∅, 𝐵) ∈ 𝒫 𝐴)
115		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝐴 ↔ 𝐵 = 𝐴))
116		id 22	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
117	115, 116	ifbieq2d 4111	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐴, ∅, 𝑥) = if(𝐵 = 𝐴, ∅, 𝐵))
118	117, 72	fvmptg 6280	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐵 ∈ 𝒫 𝐴 ∧ if(𝐵 = 𝐴, ∅, 𝐵) ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵) = if(𝐵 = 𝐴, ∅, 𝐵))
119	101, 114, 118	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵) = if(𝐵 = 𝐴, ∅, 𝐵))
120		pssne 3703	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ≠ 𝐴)
121	120	neneqd 2799	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐵 = 𝐴)
122	121	iffalsed 4097	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐵 ⊊ 𝐴 → if(𝐵 = 𝐴, ∅, 𝐵) = 𝐵)
123	119, 122	sylan9eq 2676	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵) = 𝐵)
124	123	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → ((𝐺 ∘ 𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵)) = ((𝐺 ∘ 𝐹)‘𝐵))
125		ifcl 4130	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ∈ 𝒫 𝐴) → if((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) = 𝐴, ∅, (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) ∈ 𝒫 𝐴)
126	56, 108, 125	sylancr 695	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → if((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) = 𝐴, ∅, (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) ∈ 𝒫 𝐴)
127		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) → (𝑥 = 𝐴 ↔ (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) = 𝐴))
128		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) → 𝑥 = (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}))
129	127, 128	ifbieq2d 4111	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) → if(𝑥 = 𝐴, ∅, 𝑥) = if((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) = 𝐴, ∅, (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
130	129, 72	fvmptg 6280	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ∈ 𝒫 𝐴 ∧ if((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) = 𝐴, ∅, (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) = if((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) = 𝐴, ∅, (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
131	108, 126, 130	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) = if((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) = 𝐴, ∅, (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
132	131	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) = if((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) = 𝐴, ∅, (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
133		sspsstr 3712	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐴) → (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ⊊ 𝐴)
134	104, 133	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ⊊ 𝐴)
135	134	pssned 3705	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ≠ 𝐴)
136	135	neneqd 2799	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → ¬ (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) = 𝐴)
137	136	iffalsed 4097	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → if((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) = 𝐴, ∅, (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) = (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}))
138	132, 137	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) = (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}))
139	138	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → ((𝐺 ∘ 𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}))) = ((𝐺 ∘ 𝐹)‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
140	112, 124, 139	3eqtr3d 2664	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → ((𝐺 ∘ 𝐹)‘𝐵) = ((𝐺 ∘ 𝐹)‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
141	101, 120	anim12i 590	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → (𝐵 ∈ 𝒫 𝐴 ∧ 𝐵 ≠ 𝐴))
142		eldifsn 4317	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ (𝒫 𝐴 ∖ {𝐴}) ↔ (𝐵 ∈ 𝒫 𝐴 ∧ 𝐵 ≠ 𝐴))
143	141, 142	sylibr 224	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → 𝐵 ∈ (𝒫 𝐴 ∖ {𝐴}))
144		fvres 6207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐵 ∈ (𝒫 𝐴 ∖ {𝐴}) → (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘𝐵) = ((𝐺 ∘ 𝐹)‘𝐵))
145	143, 144	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘𝐵) = ((𝐺 ∘ 𝐹)‘𝐵))
146	108	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ∈ 𝒫 𝐴)
147		eldifsn 4317	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ∈ (𝒫 𝐴 ∖ {𝐴}) ↔ ((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ∈ 𝒫 𝐴 ∧ (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ≠ 𝐴))
148	146, 135, 147	sylanbrc 698	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ∈ (𝒫 𝐴 ∖ {𝐴}))
149		fvres 6207	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ∈ (𝒫 𝐴 ∖ {𝐴}) → (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) = ((𝐺 ∘ 𝐹)‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
150	148, 149	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) = ((𝐺 ∘ 𝐹)‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
151	140, 145, 150	3eqtr4d 2666	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘𝐵) = (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
152		f1of1 6136	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→𝐴 → ((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1→𝐴)
153	53, 152	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1→𝐴)
154	153	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → ((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1→𝐴)
155		f1fveq 6519	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1→𝐴 ∧ (𝐵 ∈ (𝒫 𝐴 ∖ {𝐴}) ∧ (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) ∈ (𝒫 𝐴 ∖ {𝐴}))) → ((((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘𝐵) = (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) ↔ 𝐵 = (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
156	154, 143, 148, 155	syl12anc 1324	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → ((((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘𝐵) = (((𝐺 ∘ 𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘(◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})) ↔ 𝐵 = (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
157	151, 156	mpbid 222	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐵 ⊊ 𝐴) → 𝐵 = (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}))
158	157	ex 450	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 ⊊ 𝐴 → 𝐵 = (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)})))
159	158	necon3ad 2807	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 ≠ (◡(𝑊‘𝐵) “ {(𝐻‘𝐵)}) → ¬ 𝐵 ⊊ 𝐴))
160	94, 159	mpd 15	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ 𝐵 ⊊ 𝐴)
161		npss 3717	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 → 𝐵 = 𝐴))
162	160, 161	sylib 208	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 ⊆ 𝐴 → 𝐵 = 𝐴))
163	91, 162	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = 𝐴)
164		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐵 = 𝐵
165		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑊‘𝐵) = (𝑊‘𝐵)
166	164, 165	pm3.2i 471	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))
167	84	sseli 3599	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ dom card) → 𝑥 ∈ 𝒫 𝐴)
168		ffvelrn 6357	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐻:𝒫 𝐴⟶𝐴 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝐻‘𝑥) ∈ 𝐴)
169	83, 167, 168	syl2an 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐻‘𝑥) ∈ 𝐴)
170	86, 4, 169, 87	fpwwe 9468	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐵𝑊(𝑊‘𝐵) ∧ (𝐻‘𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))))
171	166, 170	mpbiri 248	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐵𝑊(𝑊‘𝐵) ∧ (𝐻‘𝐵) ∈ 𝐵))
172	171	simpld 475	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵𝑊(𝑊‘𝐵))
173	86, 4	fpwwelem 9467	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵𝑊(𝑊‘𝐵) ↔ ((𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐻‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦))))
174	172, 173	mpbid 222	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐻‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦)))
175	174	simprd 479	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐻‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦))
176	175	simpld 475	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑊‘𝐵) We 𝐵)
177		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (𝑊‘𝐵) ∈ V
178		weeq1 5102	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = (𝑊‘𝐵) → (𝑟 We 𝐵 ↔ (𝑊‘𝐵) We 𝐵))
179	177, 178	spcev 3300	. . . . . . . . . . . . . 14 ⊢ ((𝑊‘𝐵) We 𝐵 → ∃𝑟 𝑟 We 𝐵)
180	176, 179	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑟 𝑟 We 𝐵)
181		ween 8858	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵)
182	180, 181	sylibr 224	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ dom card)
183	163, 182	eqeltrrd 2702	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ dom card)
184		domtri2 8815	. . . . . . . . . . 11 ⊢ ((ω ∈ dom card ∧ 𝐴 ∈ dom card) → (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω))
185	21, 183, 184	sylancr 695	. . . . . . . . . 10 ⊢ (𝜑 → (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω))
186		infcda1 9015	. . . . . . . . . 10 ⊢ (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
187	185, 186	syl6bir 244	. . . . . . . . 9 ⊢ (𝜑 → (¬ 𝐴 ≺ ω → (𝐴 +𝑐 1𝑜) ≈ 𝐴))
188		ensym 8005	. . . . . . . . 9 ⊢ ((𝐴 +𝑐 1𝑜) ≈ 𝐴 → 𝐴 ≈ (𝐴 +𝑐 1𝑜))
189	187, 188	syl6 35	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝐴 ≺ ω → 𝐴 ≈ (𝐴 +𝑐 1𝑜)))
190	18, 189	mt3d 140	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≺ ω)
191		2onn 7720	. . . . . . . 8 ⊢ 2𝑜 ∈ ω
192		nnsdom 8551	. . . . . . . 8 ⊢ (2𝑜 ∈ ω → 2𝑜 ≺ ω)
193	191, 192	ax-mp 5	. . . . . . 7 ⊢ 2𝑜 ≺ ω
194		cdafi 9012	. . . . . . 7 ⊢ ((𝐴 ≺ ω ∧ 2𝑜 ≺ ω) → (𝐴 +𝑐 2𝑜) ≺ ω)
195	190, 193, 194	sylancl 694	. . . . . 6 ⊢ (𝜑 → (𝐴 +𝑐 2𝑜) ≺ ω)
196		isfinite 8549	. . . . . 6 ⊢ ((𝐴 +𝑐 2𝑜) ∈ Fin ↔ (𝐴 +𝑐 2𝑜) ≺ ω)
197	195, 196	sylibr 224	. . . . 5 ⊢ (𝜑 → (𝐴 +𝑐 2𝑜) ∈ Fin)
198		sssucid 5802	. . . . . . . . . 10 ⊢ 1𝑜 ⊆ suc 1𝑜
199		df-2o 7561	. . . . . . . . . 10 ⊢ 2𝑜 = suc 1𝑜
200	198, 199	sseqtr4i 3638	. . . . . . . . 9 ⊢ 1𝑜 ⊆ 2𝑜
201		xpss1 5228	. . . . . . . . 9 ⊢ (1𝑜 ⊆ 2𝑜 → (1𝑜 × {1𝑜}) ⊆ (2𝑜 × {1𝑜}))
202	200, 201	ax-mp 5	. . . . . . . 8 ⊢ (1𝑜 × {1𝑜}) ⊆ (2𝑜 × {1𝑜})
203		unss2 3784	. . . . . . . 8 ⊢ ((1𝑜 × {1𝑜}) ⊆ (2𝑜 × {1𝑜}) → ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊆ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})))
204	202, 203	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊆ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})))
205		ssun2 3777	. . . . . . . . 9 ⊢ (2𝑜 × {1𝑜}) ⊆ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜}))
206		1onn 7719	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ ω
207	206	elexi 3213	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ V
208	207	sucid 5804	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ suc 1𝑜
209	208, 199	eleqtrri 2700	. . . . . . . . . 10 ⊢ 1𝑜 ∈ 2𝑜
210	207	snid 4208	. . . . . . . . . 10 ⊢ 1𝑜 ∈ {1𝑜}
211		opelxpi 5148	. . . . . . . . . 10 ⊢ ((1𝑜 ∈ 2𝑜 ∧ 1𝑜 ∈ {1𝑜}) → ⟨1𝑜, 1𝑜⟩ ∈ (2𝑜 × {1𝑜}))
212	209, 210, 211	mp2an 708	. . . . . . . . 9 ⊢ ⟨1𝑜, 1𝑜⟩ ∈ (2𝑜 × {1𝑜})
213	205, 212	sselii 3600	. . . . . . . 8 ⊢ ⟨1𝑜, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜}))
214		1n0 7575	. . . . . . . . . . . 12 ⊢ 1𝑜 ≠ ∅
215	214	neii 2796	. . . . . . . . . . 11 ⊢ ¬ 1𝑜 = ∅
216		opelxp2 5151	. . . . . . . . . . . 12 ⊢ (⟨1𝑜, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 ∈ {∅})
217		elsni 4194	. . . . . . . . . . . 12 ⊢ (1𝑜 ∈ {∅} → 1𝑜 = ∅)
218	216, 217	syl 17	. . . . . . . . . . 11 ⊢ (⟨1𝑜, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 = ∅)
219	215, 218	mto 188	. . . . . . . . . 10 ⊢ ¬ ⟨1𝑜, 1𝑜⟩ ∈ (𝐴 × {∅})
220		nnord 7073	. . . . . . . . . . . 12 ⊢ (1𝑜 ∈ ω → Ord 1𝑜)
221		ordirr 5741	. . . . . . . . . . . 12 ⊢ (Ord 1𝑜 → ¬ 1𝑜 ∈ 1𝑜)
222	206, 220, 221	mp2b 10	. . . . . . . . . . 11 ⊢ ¬ 1𝑜 ∈ 1𝑜
223		opelxp1 5150	. . . . . . . . . . 11 ⊢ (⟨1𝑜, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}) → 1𝑜 ∈ 1𝑜)
224	222, 223	mto 188	. . . . . . . . . 10 ⊢ ¬ ⟨1𝑜, 1𝑜⟩ ∈ (1𝑜 × {1𝑜})
225	219, 224	pm3.2ni 899	. . . . . . . . 9 ⊢ ¬ (⟨1𝑜, 1𝑜⟩ ∈ (𝐴 × {∅}) ∨ ⟨1𝑜, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}))
226		elun 3753	. . . . . . . . 9 ⊢ (⟨1𝑜, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ↔ (⟨1𝑜, 1𝑜⟩ ∈ (𝐴 × {∅}) ∨ ⟨1𝑜, 1𝑜⟩ ∈ (1𝑜 × {1𝑜})))
227	225, 226	mtbir 313	. . . . . . . 8 ⊢ ¬ ⟨1𝑜, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
228		ssnelpss 3718	. . . . . . . 8 ⊢ (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊆ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})) → ((⟨1𝑜, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})) ∧ ¬ ⟨1𝑜, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))) → ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊊ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜}))))
229	213, 227, 228	mp2ani 714	. . . . . . 7 ⊢ (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊆ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})) → ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊊ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})))
230	204, 229	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊊ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})))
231		cdaval 8992	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ∈ ω) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
232	4, 206, 231	sylancl 694	. . . . . . 7 ⊢ (𝜑 → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
233		cdaval 8992	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 2𝑜 ∈ ω) → (𝐴 +𝑐 2𝑜) = ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})))
234	4, 191, 233	sylancl 694	. . . . . . 7 ⊢ (𝜑 → (𝐴 +𝑐 2𝑜) = ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})))
235	232, 234	psseq12d 3701	. . . . . 6 ⊢ (𝜑 → ((𝐴 +𝑐 1𝑜) ⊊ (𝐴 +𝑐 2𝑜) ↔ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊊ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜}))))
236	230, 235	mpbird 247	. . . . 5 ⊢ (𝜑 → (𝐴 +𝑐 1𝑜) ⊊ (𝐴 +𝑐 2𝑜))
237		php3 8146	. . . . 5 ⊢ (((𝐴 +𝑐 2𝑜) ∈ Fin ∧ (𝐴 +𝑐 1𝑜) ⊊ (𝐴 +𝑐 2𝑜)) → (𝐴 +𝑐 1𝑜) ≺ (𝐴 +𝑐 2𝑜))
238	197, 236, 237	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐴 +𝑐 1𝑜) ≺ (𝐴 +𝑐 2𝑜))
239		canthp1lem1 9474	. . . . 5 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 +𝑐 2𝑜) ≼ 𝒫 𝐴)
240	1, 239	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 +𝑐 2𝑜) ≼ 𝒫 𝐴)
241		sdomdomtr 8093	. . . 4 ⊢ (((𝐴 +𝑐 1𝑜) ≺ (𝐴 +𝑐 2𝑜) ∧ (𝐴 +𝑐 2𝑜) ≼ 𝒫 𝐴) → (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴)
242	238, 240, 241	syl2anc 693	. . 3 ⊢ (𝜑 → (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴)
243		sdomnen 7984	. . 3 ⊢ ((𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴 → ¬ (𝐴 +𝑐 1𝑜) ≈ 𝒫 𝐴)
244	242, 243	syl 17	. 2 ⊢ (𝜑 → ¬ (𝐴 +𝑐 1𝑜) ≈ 𝒫 𝐴)
245	11, 244	pm2.65i 185	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574   ⊊ wpss 3575  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  {csn 4177  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653  {copab 4712   ↦ cmpt 4729   We wwe 5072   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117   ∘ ccom 5118  Ord word 5722  Oncon0 5723  suc csuc 5725  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  ωcom 7065  1_𝑜c1o 7553  2_𝑜c2o 7554   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954  Fincfn 7955  cardccrd 8761   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-cda 8990

This theorem is referenced by:  canthp1  9476