Theorem oemapvali 8581

Description: If 𝐹 < 𝐺, then there is some 𝑧 witnessing this, but we can say more and in fact there is a definable expression 𝑋 that also witnesses 𝐹 < 𝐺. (Contributed by Mario Carneiro, 25-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
oemapval.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
oemapval.f	⊢ (𝜑 → 𝐹 ∈ 𝑆)
oemapval.g	⊢ (𝜑 → 𝐺 ∈ 𝑆)
oemapvali.r	⊢ (𝜑 → 𝐹𝑇𝐺)
oemapvali.x	⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}

Assertion

Ref	Expression
oemapvali	⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))

Distinct variable groups:   𝑤,𝑐,𝑥,𝑦,𝑧,𝐵   𝐴,𝑐,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐   𝑤,𝐹,𝑥,𝑦,𝑧   𝑆,𝑐,𝑥,𝑦,𝑧   𝐺,𝑐,𝑤,𝑥,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑤,𝑋,𝑥,𝑦,𝑧   𝐹,𝑐   𝜑,𝑐

Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝑋(𝑐)

Proof of Theorem oemapvali

Step	Hyp	Ref	Expression
1		oemapvali.r	. . 3 ⊢ (𝜑 → 𝐹𝑇𝐺)
2		cantnfs.s	. . . 4 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
3		cantnfs.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ On)
4		cantnfs.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ On)
5		oemapval.t	. . . 4 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
6		oemapval.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
7		oemapval.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑆)
8	2, 3, 4, 5, 6, 7	oemapval 8580	. . 3 ⊢ (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))))
9	1, 8	mpbid 222	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
10		ssrab2 3687	. . . 4 ⊢ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ 𝐵
11		oemapvali.x	. . . . 5 ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}
12	4	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐵 ∈ On)
13		onss 6990	. . . . . . . 8 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
14	12, 13	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐵 ⊆ On)
15	10, 14	syl5ss 3614	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ On)
16	2, 3, 4	cantnfs 8563	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
17	7, 16	mpbid 222	. . . . . . . . 9 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
18	17	simprd 479	. . . . . . . 8 ⊢ (𝜑 → 𝐺 finSupp ∅)
19	18	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐺 finSupp ∅)
20	4	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → 𝐵 ∈ On)
21		simp2 1062	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → 𝑐 ∈ 𝐵)
22	17	simpld 475	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
23		ffn 6045	. . . . . . . . . . . 12 ⊢ (𝐺:𝐵⟶𝐴 → 𝐺 Fn 𝐵)
24	22, 23	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 Fn 𝐵)
25	24	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → 𝐺 Fn 𝐵)
26		ne0i 3921	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑐) ∈ (𝐺‘𝑐) → (𝐺‘𝑐) ≠ ∅)
27	26	3ad2ant3 1084	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → (𝐺‘𝑐) ≠ ∅)
28		fvn0elsupp 7311	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝑐 ∈ 𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺‘𝑐) ≠ ∅)) → 𝑐 ∈ (𝐺 supp ∅))
29	20, 21, 25, 27, 28	syl22anc 1327	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → 𝑐 ∈ (𝐺 supp ∅))
30	29	rabssdv 3682	. . . . . . . 8 ⊢ (𝜑 → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅))
31	30	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅))
32		fsuppimp 8281	. . . . . . . 8 ⊢ (𝐺 finSupp ∅ → (Fun 𝐺 ∧ (𝐺 supp ∅) ∈ Fin))
33		ssfi 8180	. . . . . . . . . 10 ⊢ (((𝐺 supp ∅) ∈ Fin ∧ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅)) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin)
34	33	ex 450	. . . . . . . . 9 ⊢ ((𝐺 supp ∅) ∈ Fin → ({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin))
35	34	adantl 482	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ (𝐺 supp ∅) ∈ Fin) → ({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin))
36	32, 35	syl 17	. . . . . . 7 ⊢ (𝐺 finSupp ∅ → ({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin))
37	19, 31, 36	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin)
38		simprl 794	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑧 ∈ 𝐵)
39		simprrl 804	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝐹‘𝑧) ∈ (𝐺‘𝑧))
40		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (𝐹‘𝑐) = (𝐹‘𝑧))
41		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (𝐺‘𝑐) = (𝐺‘𝑧))
42	40, 41	eleq12d 2695	. . . . . . . . 9 ⊢ (𝑐 = 𝑧 → ((𝐹‘𝑐) ∈ (𝐺‘𝑐) ↔ (𝐹‘𝑧) ∈ (𝐺‘𝑧)))
43	42	elrab 3363	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ↔ (𝑧 ∈ 𝐵 ∧ (𝐹‘𝑧) ∈ (𝐺‘𝑧)))
44	38, 39, 43	sylanbrc 698	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑧 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)})
45		ne0i 3921	. . . . . . 7 ⊢ (𝑧 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ≠ ∅)
46	44, 45	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ≠ ∅)
47		ordunifi 8210	. . . . . 6 ⊢ (({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ On ∧ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin ∧ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ≠ ∅) → ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)})
48	15, 37, 46, 47	syl3anc 1326	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)})
49	11, 48	syl5eqel 2705	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)})
50	10, 49	sseldi 3601	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 ∈ 𝐵)
51		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
52		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
53	51, 52	eleq12d 2695	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ∈ (𝐺‘𝑥) ↔ (𝐹‘𝑋) ∈ (𝐺‘𝑋)))
54		fveq2 6191	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (𝐹‘𝑐) = (𝐹‘𝑥))
55		fveq2 6191	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (𝐺‘𝑐) = (𝐺‘𝑥))
56	54, 55	eleq12d 2695	. . . . . . 7 ⊢ (𝑐 = 𝑥 → ((𝐹‘𝑐) ∈ (𝐺‘𝑐) ↔ (𝐹‘𝑥) ∈ (𝐺‘𝑥)))
57	56	cbvrabv 3199	. . . . . 6 ⊢ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} = {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) ∈ (𝐺‘𝑥)}
58	53, 57	elrab2 3366	. . . . 5 ⊢ (𝑋 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ↔ (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋)))
59	49, 58	sylib 208	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋)))
60	59	simprd 479	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝐹‘𝑋) ∈ (𝐺‘𝑋))
61		simprrr 805	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))
62	3	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐴 ∈ On)
63	22	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐺:𝐵⟶𝐴)
64	63, 50	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝐺‘𝑋) ∈ 𝐴)
65		onelon 5748	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑋) ∈ 𝐴) → (𝐺‘𝑋) ∈ On)
66	62, 64, 65	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝐺‘𝑋) ∈ On)
67		eloni 5733	. . . . . . . . . 10 ⊢ ((𝐺‘𝑋) ∈ On → Ord (𝐺‘𝑋))
68		ordirr 5741	. . . . . . . . . 10 ⊢ (Ord (𝐺‘𝑋) → ¬ (𝐺‘𝑋) ∈ (𝐺‘𝑋))
69	66, 67, 68	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ¬ (𝐺‘𝑋) ∈ (𝐺‘𝑋))
70		nelneq 2725	. . . . . . . . 9 ⊢ (((𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ¬ (𝐺‘𝑋) ∈ (𝐺‘𝑋)) → ¬ (𝐹‘𝑋) = (𝐺‘𝑋))
71	60, 69, 70	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ¬ (𝐹‘𝑋) = (𝐺‘𝑋))
72		eleq2 2690	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑋 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑋))
73		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑋 → (𝐹‘𝑤) = (𝐹‘𝑋))
74		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑋 → (𝐺‘𝑤) = (𝐺‘𝑋))
75	73, 74	eqeq12d 2637	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑋 → ((𝐹‘𝑤) = (𝐺‘𝑤) ↔ (𝐹‘𝑋) = (𝐺‘𝑋)))
76	72, 75	imbi12d 334	. . . . . . . . . 10 ⊢ (𝑤 = 𝑋 → ((𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) ↔ (𝑧 ∈ 𝑋 → (𝐹‘𝑋) = (𝐺‘𝑋))))
77	76	rspccv 3306	. . . . . . . . 9 ⊢ (∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) → (𝑋 ∈ 𝐵 → (𝑧 ∈ 𝑋 → (𝐹‘𝑋) = (𝐺‘𝑋))))
78	61, 50, 77	sylc 65	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝑧 ∈ 𝑋 → (𝐹‘𝑋) = (𝐺‘𝑋)))
79	71, 78	mtod 189	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ¬ 𝑧 ∈ 𝑋)
80		ssexg 4804	. . . . . . . . . . 11 ⊢ (({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ 𝐵 ∧ 𝐵 ∈ On) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ V)
81	10, 12, 80	sylancr 695	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ V)
82		ssonuni 6986	. . . . . . . . . 10 ⊢ ({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ V → ({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ On → ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ On))
83	81, 15, 82	sylc 65	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ On)
84	11, 83	syl5eqel 2705	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 ∈ On)
85		onelon 5748	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ On)
86	12, 38, 85	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑧 ∈ On)
87		ontri1 5757	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑧 ∈ On) → (𝑋 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑋))
88	84, 86, 87	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝑋 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑋))
89	79, 88	mpbird 247	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 ⊆ 𝑧)
90		elssuni 4467	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} → 𝑧 ⊆ ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)})
91	90, 11	syl6sseqr 3652	. . . . . . 7 ⊢ (𝑧 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} → 𝑧 ⊆ 𝑋)
92	44, 91	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑧 ⊆ 𝑋)
93	89, 92	eqssd 3620	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 = 𝑧)
94		eleq1 2689	. . . . . . 7 ⊢ (𝑋 = 𝑧 → (𝑋 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤))
95	94	imbi1d 331	. . . . . 6 ⊢ (𝑋 = 𝑧 → ((𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) ↔ (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
96	95	ralbidv 2986	. . . . 5 ⊢ (𝑋 = 𝑧 → (∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) ↔ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
97	93, 96	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) ↔ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
98	61, 97	mpbird 247	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))
99	50, 60, 98	3jca 1242	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
100	9, 99	rexlimddv 3035	1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ∪ cuni 4436   class class class wbr 4653  {copab 4712  dom cdm 5114  Ord word 5722  Oncon0 5723  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   supp csupp 7295  Fincfn 7955   finSupp cfsupp 8275   CNF ccnf 8558

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-supp 7296  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-fin 7959  df-fsupp 8276  df-cnf 8559

This theorem is referenced by:  cantnflem1a  8582  cantnflem1b  8583  cantnflem1c  8584  cantnflem1d  8585  cantnflem1  8586