Theorem pwfseqlem5 9485

Description: Lemma for pwfseq 9486. Although in some ways pwfseqlem4 9484 is the "main" part of the proof, one last aspect which makes up a remark in the original text is by far the hardest part to formalize. The main proof relies on the existence of an injection 𝐾 from the set of finite sequences on an infinite set 𝑥 to 𝑥. Now this alone would not be difficult to prove; this is mostly the claim of fseqen 8850. However, what is needed for the proof is a canonical injection on these sets, so we have to start from scratch pulling together explicit bijections from the lemmas.

If one attempts such a program, it will mostly go through, but there is one key step which is inherently nonconstructive, namely the proof of infxpen 8837. The resolution is not obvious, but it turns out that reversing an infinite ordinal's Cantor normal form absorbs all the non-leading terms (cnfcom3c 8603), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 8841). (Contributed by Mario Carneiro, 31-May-2015.)

Hypotheses

Ref	Expression
pwfseqlem5.g	⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
pwfseqlem5.x	⊢ (𝜑 → 𝑋 ⊆ 𝐴)
pwfseqlem5.h	⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)
pwfseqlem5.ps	⊢ (𝜓 ↔ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡))
pwfseqlem5.n	⊢ (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
pwfseqlem5.o	⊢ 𝑂 = OrdIso(𝑟, 𝑡)
pwfseqlem5.t	⊢ 𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩)
pwfseqlem5.p	⊢ 𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ◡𝑇)
pwfseqlem5.s	⊢ 𝑆 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑡 ↑𝑚 suc 𝑘) ↦ ((𝑓‘(𝑥 ↾ 𝑘))𝑃(𝑥‘𝑘)))), {⟨∅, (𝑂‘∅)⟩})
pwfseqlem5.q	⊢ 𝑄 = (𝑦 ∈ ∪ 𝑛 ∈ ω (𝑡 ↑𝑚 𝑛) ↦ ⟨dom 𝑦, ((𝑆‘dom 𝑦)‘𝑦)⟩)
pwfseqlem5.i	⊢ 𝐼 = (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩)
pwfseqlem5.k	⊢ 𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄)

Assertion

Ref	Expression
pwfseqlem5	⊢ ¬ 𝜑

Distinct variable groups:   𝑛,𝑏,𝐺   𝑟,𝑏,𝑡,𝐻   𝑓,𝑘,𝑥,𝑃   𝑓,𝑏,𝑘,𝑢,𝑣,𝑥,𝑦,𝑛,𝑟,𝑡   𝜑,𝑏,𝑘,𝑛,𝑟,𝑡,𝑥,𝑦   𝐾,𝑏,𝑛   𝑁,𝑏   𝜓,𝑘,𝑛,𝑥,𝑦   𝑆,𝑛,𝑦   𝐴,𝑏,𝑛,𝑟,𝑡   𝑂,𝑏,𝑢,𝑣,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑣,𝑢,𝑓)   𝜓(𝑣,𝑢,𝑡,𝑓,𝑟,𝑏)   𝐴(𝑥,𝑦,𝑣,𝑢,𝑓,𝑘)   𝑃(𝑦,𝑣,𝑢,𝑡,𝑛,𝑟,𝑏)   𝑄(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑛,𝑟,𝑏)   𝑆(𝑥,𝑣,𝑢,𝑡,𝑓,𝑘,𝑟,𝑏)   𝑇(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑛,𝑟,𝑏)   𝐺(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑟)   𝐻(𝑥,𝑦,𝑣,𝑢,𝑓,𝑘,𝑛)   𝐼(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑛,𝑟,𝑏)   𝐾(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑟)   𝑁(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑛,𝑟)   𝑂(𝑡,𝑓,𝑘,𝑛,𝑟)   𝑋(𝑥,𝑦,𝑣,𝑢,𝑡,𝑓,𝑘,𝑛,𝑟,𝑏)

Proof of Theorem pwfseqlem5

Dummy variables 𝑎 𝑐 𝑑 𝑖 𝑗 𝑚 𝑠 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwfseqlem5.g	. 2 ⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
2		pwfseqlem5.x	. 2 ⊢ (𝜑 → 𝑋 ⊆ 𝐴)
3		pwfseqlem5.h	. 2 ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)
4		pwfseqlem5.ps	. 2 ⊢ (𝜓 ↔ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡))
5		vex 3203	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
6		simprl3 1108	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑟 We 𝑡)
7	4, 6	sylan2b 492	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝑟 We 𝑡)
8		pwfseqlem5.o	. . . . . . . . . . . 12 ⊢ 𝑂 = OrdIso(𝑟, 𝑡)
9	8	oiiso 8442	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
10	5, 7, 9	sylancr 695	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡))
11		isof1o 6573	. . . . . . . . . 10 ⊢ (𝑂 Isom E , 𝑟 (dom 𝑂, 𝑡) → 𝑂:dom 𝑂–1-1-onto→𝑡)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑂:dom 𝑂–1-1-onto→𝑡)
13	8	oion 8441	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ V → dom 𝑂 ∈ On)
14	5, 13	ax-mp 5	. . . . . . . . . . . 12 ⊢ dom 𝑂 ∈ On
15	14	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ On)
16	8	oien 8443	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ V ∧ 𝑟 We 𝑡) → dom 𝑂 ≈ 𝑡)
17	5, 7, 16	sylancr 695	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ≈ 𝑡)
18	1	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
19		omex 8540	. . . . . . . . . . . . . . . . 17 ⊢ ω ∈ V
20		ovex 6678	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ↑𝑚 𝑛) ∈ V
21	19, 20	iunex 7147	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) ∈ V
22		f1dmex 7136	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) ∧ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) ∈ V) → 𝒫 𝐴 ∈ V)
23	18, 21, 22	sylancl 694	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝒫 𝐴 ∈ V)
24		pwexb 6975	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
25	23, 24	sylibr 224	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ V)
26		simprl1 1106	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → 𝑡 ⊆ 𝐴)
27	4, 26	sylan2b 492	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ⊆ 𝐴)
28		ssdomg 8001	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ V → (𝑡 ⊆ 𝐴 → 𝑡 ≼ 𝐴))
29	25, 27, 28	sylc 65	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≼ 𝐴)
30		canth2g 8114	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
31		sdomdom 7983	. . . . . . . . . . . . . 14 ⊢ (𝐴 ≺ 𝒫 𝐴 → 𝐴 ≼ 𝒫 𝐴)
32	25, 30, 31	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ≼ 𝒫 𝐴)
33		domtr 8009	. . . . . . . . . . . . 13 ⊢ ((𝑡 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐴) → 𝑡 ≼ 𝒫 𝐴)
34	29, 32, 33	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≼ 𝒫 𝐴)
35		endomtr 8014	. . . . . . . . . . . 12 ⊢ ((dom 𝑂 ≈ 𝑡 ∧ 𝑡 ≼ 𝒫 𝐴) → dom 𝑂 ≼ 𝒫 𝐴)
36	17, 34, 35	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ≼ 𝒫 𝐴)
37		elharval 8468	. . . . . . . . . . 11 ⊢ (dom 𝑂 ∈ (har‘𝒫 𝐴) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐴))
38	15, 36, 37	sylanbrc 698	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ (har‘𝒫 𝐴))
39		pwfseqlem5.n	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
40	39	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
41		cardom 8812	. . . . . . . . . . . 12 ⊢ (card‘ω) = ω
42		simprr 796	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) → ω ≼ 𝑡)
43	4, 42	sylan2b 492	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ 𝑡)
44	17	ensymd 8007	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ≈ dom 𝑂)
45		domentr 8015	. . . . . . . . . . . . . 14 ⊢ ((ω ≼ 𝑡 ∧ 𝑡 ≈ dom 𝑂) → ω ≼ dom 𝑂)
46	43, 44, 45	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ dom 𝑂)
47		omelon 8543	. . . . . . . . . . . . . . 15 ⊢ ω ∈ On
48		onenon 8775	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ On → ω ∈ dom card)
49	47, 48	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ω ∈ dom card
50		onenon 8775	. . . . . . . . . . . . . . 15 ⊢ (dom 𝑂 ∈ On → dom 𝑂 ∈ dom card)
51	14, 50	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑂 ∈ dom card)
52		carddom2 8803	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ dom card ∧ dom 𝑂 ∈ dom card) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
53	49, 51, 52	sylancr 695	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((card‘ω) ⊆ (card‘dom 𝑂) ↔ ω ≼ dom 𝑂))
54	46, 53	mpbird 247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (card‘ω) ⊆ (card‘dom 𝑂))
55	41, 54	syl5eqssr 3650	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ω ⊆ (card‘dom 𝑂))
56		cardonle 8783	. . . . . . . . . . . 12 ⊢ (dom 𝑂 ∈ On → (card‘dom 𝑂) ⊆ dom 𝑂)
57	15, 56	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (card‘dom 𝑂) ⊆ dom 𝑂)
58	55, 57	sstrd 3613	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ω ⊆ dom 𝑂)
59		sseq2 3627	. . . . . . . . . . . 12 ⊢ (𝑏 = dom 𝑂 → (ω ⊆ 𝑏 ↔ ω ⊆ dom 𝑂))
60		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑏 = dom 𝑂 → (𝑁‘𝑏) = (𝑁‘dom 𝑂))
61		f1oeq1 6127	. . . . . . . . . . . . . 14 ⊢ ((𝑁‘𝑏) = (𝑁‘dom 𝑂) → ((𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto→𝑏))
62	60, 61	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto→𝑏))
63		xpeq12 5134	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = dom 𝑂 ∧ 𝑏 = dom 𝑂) → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
64	63	anidms 677	. . . . . . . . . . . . . 14 ⊢ (𝑏 = dom 𝑂 → (𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂))
65		f1oeq2 6128	. . . . . . . . . . . . . 14 ⊢ ((𝑏 × 𝑏) = (dom 𝑂 × dom 𝑂) → ((𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→𝑏))
66	64, 65	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘dom 𝑂):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→𝑏))
67		f1oeq3 6129	. . . . . . . . . . . . 13 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂))
68	62, 66, 67	3bitrd 294	. . . . . . . . . . . 12 ⊢ (𝑏 = dom 𝑂 → ((𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂))
69	59, 68	imbi12d 334	. . . . . . . . . . 11 ⊢ (𝑏 = dom 𝑂 → ((ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏) ↔ (ω ⊆ dom 𝑂 → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂)))
70	69	rspcv 3305	. . . . . . . . . 10 ⊢ (dom 𝑂 ∈ (har‘𝒫 𝐴) → (∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏) → (ω ⊆ dom 𝑂 → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂)))
71	38, 40, 58, 70	syl3c 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂)
72		f1oco 6159	. . . . . . . . 9 ⊢ ((𝑂:dom 𝑂–1-1-onto→𝑡 ∧ (𝑁‘dom 𝑂):(dom 𝑂 × dom 𝑂)–1-1-onto→dom 𝑂) → (𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto→𝑡)
73	12, 71, 72	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto→𝑡)
74		f1of 6137	. . . . . . . . . . . . . . 15 ⊢ (𝑂:dom 𝑂–1-1-onto→𝑡 → 𝑂:dom 𝑂⟶𝑡)
75	12, 74	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑂:dom 𝑂⟶𝑡)
76	75	feqmptd 6249	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 = (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)))
77		f1oeq1 6127	. . . . . . . . . . . . 13 ⊢ (𝑂 = (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)) → (𝑂:dom 𝑂–1-1-onto→𝑡 ↔ (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)):dom 𝑂–1-1-onto→𝑡))
78	76, 77	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑂:dom 𝑂–1-1-onto→𝑡 ↔ (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)):dom 𝑂–1-1-onto→𝑡))
79	12, 78	mpbid 222	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑢 ∈ dom 𝑂 ↦ (𝑂‘𝑢)):dom 𝑂–1-1-onto→𝑡)
80	75	feqmptd 6249	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑂 = (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)))
81		f1oeq1 6127	. . . . . . . . . . . . 13 ⊢ (𝑂 = (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)) → (𝑂:dom 𝑂–1-1-onto→𝑡 ↔ (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)):dom 𝑂–1-1-onto→𝑡))
82	80, 81	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑂:dom 𝑂–1-1-onto→𝑡 ↔ (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)):dom 𝑂–1-1-onto→𝑡))
83	12, 82	mpbid 222	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑣 ∈ dom 𝑂 ↦ (𝑂‘𝑣)):dom 𝑂–1-1-onto→𝑡)
84	79, 83	xpf1o 8122	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩):(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
85		pwfseqlem5.t	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩)
86		f1oeq1 6127	. . . . . . . . . . 11 ⊢ (𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩) → (𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡) ↔ (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩):(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡)))
87	85, 86	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡) ↔ (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩):(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
88	84, 87	sylibr 224	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡))
89		f1ocnv 6149	. . . . . . . . 9 ⊢ (𝑇:(dom 𝑂 × dom 𝑂)–1-1-onto→(𝑡 × 𝑡) → ◡𝑇:(𝑡 × 𝑡)–1-1-onto→(dom 𝑂 × dom 𝑂))
90	88, 89	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ◡𝑇:(𝑡 × 𝑡)–1-1-onto→(dom 𝑂 × dom 𝑂))
91		f1oco 6159	. . . . . . . 8 ⊢ (((𝑂 ∘ (𝑁‘dom 𝑂)):(dom 𝑂 × dom 𝑂)–1-1-onto→𝑡 ∧ ◡𝑇:(𝑡 × 𝑡)–1-1-onto→(dom 𝑂 × dom 𝑂)) → ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ◡𝑇):(𝑡 × 𝑡)–1-1-onto→𝑡)
92	73, 90, 91	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ◡𝑇):(𝑡 × 𝑡)–1-1-onto→𝑡)
93		pwfseqlem5.p	. . . . . . . 8 ⊢ 𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ◡𝑇)
94		f1oeq1 6127	. . . . . . . 8 ⊢ (𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ◡𝑇) → (𝑃:(𝑡 × 𝑡)–1-1-onto→𝑡 ↔ ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ◡𝑇):(𝑡 × 𝑡)–1-1-onto→𝑡))
95	93, 94	ax-mp 5	. . . . . . 7 ⊢ (𝑃:(𝑡 × 𝑡)–1-1-onto→𝑡 ↔ ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ◡𝑇):(𝑡 × 𝑡)–1-1-onto→𝑡)
96	92, 95	sylibr 224	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑃:(𝑡 × 𝑡)–1-1-onto→𝑡)
97		f1of1 6136	. . . . . 6 ⊢ (𝑃:(𝑡 × 𝑡)–1-1-onto→𝑡 → 𝑃:(𝑡 × 𝑡)–1-1→𝑡)
98	96, 97	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑃:(𝑡 × 𝑡)–1-1→𝑡)
99		f1of1 6136	. . . . . . . . . . . . 13 ⊢ (𝑂:dom 𝑂–1-1-onto→𝑡 → 𝑂:dom 𝑂–1-1→𝑡)
100	12, 99	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝑂:dom 𝑂–1-1→𝑡)
101		f1ssres 6108	. . . . . . . . . . . 12 ⊢ ((𝑂:dom 𝑂–1-1→𝑡 ∧ ω ⊆ dom 𝑂) → (𝑂 ↾ ω):ω–1-1→𝑡)
102	100, 58, 101	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω):ω–1-1→𝑡)
103		f1f1orn 6148	. . . . . . . . . . 11 ⊢ ((𝑂 ↾ ω):ω–1-1→𝑡 → (𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω))
104	102, 103	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω))
105	75, 58	feqresmpt 6250	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 ↾ ω) = (𝑥 ∈ ω ↦ (𝑂‘𝑥)))
106		f1oeq1 6127	. . . . . . . . . . 11 ⊢ ((𝑂 ↾ ω) = (𝑥 ∈ ω ↦ (𝑂‘𝑥)) → ((𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω) ↔ (𝑥 ∈ ω ↦ (𝑂‘𝑥)):ω–1-1-onto→ran (𝑂 ↾ ω)))
107	105, 106	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ((𝑂 ↾ ω):ω–1-1-onto→ran (𝑂 ↾ ω) ↔ (𝑥 ∈ ω ↦ (𝑂‘𝑥)):ω–1-1-onto→ran (𝑂 ↾ ω)))
108	104, 107	mpbid 222	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 ∈ ω ↦ (𝑂‘𝑥)):ω–1-1-onto→ran (𝑂 ↾ ω))
109		mptresid 5456	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑡 ↦ 𝑦) = ( I ↾ 𝑡)
110		f1oi 6174	. . . . . . . . . . 11 ⊢ ( I ↾ 𝑡):𝑡–1-1-onto→𝑡
111		f1oeq1 6127	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑡 ↦ 𝑦) = ( I ↾ 𝑡) → ((𝑦 ∈ 𝑡 ↦ 𝑦):𝑡–1-1-onto→𝑡 ↔ ( I ↾ 𝑡):𝑡–1-1-onto→𝑡))
112	110, 111	mpbiri 248	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑡 ↦ 𝑦) = ( I ↾ 𝑡) → (𝑦 ∈ 𝑡 ↦ 𝑦):𝑡–1-1-onto→𝑡)
113	109, 112	mp1i 13	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑦 ∈ 𝑡 ↦ 𝑦):𝑡–1-1-onto→𝑡)
114	108, 113	xpf1o 8122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩):(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡))
115		pwfseqlem5.i	. . . . . . . . 9 ⊢ 𝐼 = (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩)
116		f1oeq1 6127	. . . . . . . . 9 ⊢ (𝐼 = (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩) → (𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡) ↔ (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩):(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡)))
117	115, 116	ax-mp 5	. . . . . . . 8 ⊢ (𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡) ↔ (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩):(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡))
118	114, 117	sylibr 224	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡))
119		f1of1 6136	. . . . . . 7 ⊢ (𝐼:(ω × 𝑡)–1-1-onto→(ran (𝑂 ↾ ω) × 𝑡) → 𝐼:(ω × 𝑡)–1-1→(ran (𝑂 ↾ ω) × 𝑡))
120	118, 119	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐼:(ω × 𝑡)–1-1→(ran (𝑂 ↾ ω) × 𝑡))
121		f1f 6101	. . . . . . 7 ⊢ ((𝑂 ↾ ω):ω–1-1→𝑡 → (𝑂 ↾ ω):ω⟶𝑡)
122		frn 6053	. . . . . . 7 ⊢ ((𝑂 ↾ ω):ω⟶𝑡 → ran (𝑂 ↾ ω) ⊆ 𝑡)
123		xpss1 5228	. . . . . . 7 ⊢ (ran (𝑂 ↾ ω) ⊆ 𝑡 → (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡))
124	102, 121, 122, 123	4syl 19	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡))
125		f1ss 6106	. . . . . 6 ⊢ ((𝐼:(ω × 𝑡)–1-1→(ran (𝑂 ↾ ω) × 𝑡) ∧ (ran (𝑂 ↾ ω) × 𝑡) ⊆ (𝑡 × 𝑡)) → 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡))
126	120, 124, 125	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡))
127		f1co 6110	. . . . 5 ⊢ ((𝑃:(𝑡 × 𝑡)–1-1→𝑡 ∧ 𝐼:(ω × 𝑡)–1-1→(𝑡 × 𝑡)) → (𝑃 ∘ 𝐼):(ω × 𝑡)–1-1→𝑡)
128	98, 126, 127	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑃 ∘ 𝐼):(ω × 𝑡)–1-1→𝑡)
129	5	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑡 ∈ V)
130		peano1 7085	. . . . . . . 8 ⊢ ∅ ∈ ω
131	130	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ∅ ∈ ω)
132	58, 131	sseldd 3604	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ∅ ∈ dom 𝑂)
133	75, 132	ffvelrnd 6360	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑂‘∅) ∈ 𝑡)
134		pwfseqlem5.s	. . . . 5 ⊢ 𝑆 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑡 ↑𝑚 suc 𝑘) ↦ ((𝑓‘(𝑥 ↾ 𝑘))𝑃(𝑥‘𝑘)))), {⟨∅, (𝑂‘∅)⟩})
135		pwfseqlem5.q	. . . . 5 ⊢ 𝑄 = (𝑦 ∈ ∪ 𝑛 ∈ ω (𝑡 ↑𝑚 𝑛) ↦ ⟨dom 𝑦, ((𝑆‘dom 𝑦)‘𝑦)⟩)
136	129, 133, 96, 134, 135	fseqenlem2 8848	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑄:∪ 𝑛 ∈ ω (𝑡 ↑𝑚 𝑛)–1-1→(ω × 𝑡))
137		f1co 6110	. . . 4 ⊢ (((𝑃 ∘ 𝐼):(ω × 𝑡)–1-1→𝑡 ∧ 𝑄:∪ 𝑛 ∈ ω (𝑡 ↑𝑚 𝑛)–1-1→(ω × 𝑡)) → ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑𝑚 𝑛)–1-1→𝑡)
138	128, 136, 137	syl2anc 693	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑𝑚 𝑛)–1-1→𝑡)
139		pwfseqlem5.k	. . . 4 ⊢ 𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄)
140		f1eq1 6096	. . . 4 ⊢ (𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄) → (𝐾:∪ 𝑛 ∈ ω (𝑡 ↑𝑚 𝑛)–1-1→𝑡 ↔ ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑𝑚 𝑛)–1-1→𝑡))
141	139, 140	ax-mp 5	. . 3 ⊢ (𝐾:∪ 𝑛 ∈ ω (𝑡 ↑𝑚 𝑛)–1-1→𝑡 ↔ ((𝑃 ∘ 𝐼) ∘ 𝑄):∪ 𝑛 ∈ ω (𝑡 ↑𝑚 𝑛)–1-1→𝑡)
142	138, 141	sylibr 224	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑡 ↑𝑚 𝑛)–1-1→𝑡)
143		eqid 2622	. 2 ⊢ (𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))}) = (𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})
144		eqid 2622	. 2 ⊢ (𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡}))) = (𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))
145		eqid 2622	. . 3 ⊢ {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))} = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))}
146	145	fpwwe2cbv 9452	. 2 ⊢ {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))} = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏 ∈ 𝑎 [(◡𝑠 “ {𝑏}) / 𝑤](𝑤(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑠 ∩ (𝑤 × 𝑤))) = 𝑏))}
147		eqid 2622	. 2 ⊢ ∪ dom {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))} = ∪ dom {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ⊆ 𝐴 ∧ 𝑑 ⊆ (𝑐 × 𝑐)) ∧ (𝑑 We 𝑐 ∧ ∀𝑚 ∈ 𝑐 [(◡𝑑 “ {𝑚}) / 𝑗](𝑗(𝑡 ∈ V, 𝑟 ∈ V ↦ if(𝑡 ∈ Fin, (𝐻‘(card‘𝑡)), ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘∩ {𝑧 ∈ ω ∣ ¬ ((𝐺‘{𝑖 ∈ 𝑡 ∣ ((◡𝐾‘𝑖) ∈ ran 𝐺 ∧ ¬ 𝑖 ∈ (◡𝐺‘(◡𝐾‘𝑖)))})‘𝑧) ∈ 𝑡})))(𝑑 ∩ (𝑗 × 𝑗))) = 𝑚))}
148	1, 2, 3, 4, 142, 143, 144, 146, 147	pwfseqlem4 9484	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200  [wsbc 3435   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  {csn 4177  ⟨cop 4183  ∪ cuni 4436  ∩ cint 4475  ∪ ciun 4520   class class class wbr 4653  {copab 4712   ↦ cmpt 4729   I cid 5023   E cep 5028   We wwe 5072   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117   ∘ ccom 5118  Oncon0 5723  suc csuc 5725  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888   Isom wiso 5889  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065  seq_𝜔cseqom 7542   ↑_𝑚 cmap 7857   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954  Fincfn 7955  OrdIsocoi 8414  harchar 8461  cardccrd 8761

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-seqom 7543  df-1o 7560  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765

This theorem is referenced by:  pwfseq  9486