Theorem oemapvali 9666

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 9665	. . 3 ⊢ (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))))
9	1, 8	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
10		ssrab2 4028	. . . 4 ⊢ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ 𝐵
11		oemapvali.x	. . . . 5 ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}
12	4	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐵 ∈ On)
13		onss 7785	. . . . . . . 8 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
14	12, 13	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐵 ⊆ On)
15	10, 14	sstrid 3942	. . . . . 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 9648	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
17	7, 16	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
18	17	simprd 501	. . . . . . . 8 ⊢ (𝜑 → 𝐺 finSupp ∅)
19	18	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐺 finSupp ∅)
20	4	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → 𝐵 ∈ On)
21		simp2 1155	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → 𝑐 ∈ 𝐵)
22	17	simpld 500	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
23	22	ffnd 6704	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 Fn 𝐵)
24	23	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → 𝐺 Fn 𝐵)
25		ne0i 4287	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑐) ∈ (𝐺‘𝑐) → (𝐺‘𝑐) ≠ ∅)
26	25	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → (𝐺‘𝑐) ≠ ∅)
27		fvn0elsupp 8179	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝑐 ∈ 𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺‘𝑐) ≠ ∅)) → 𝑐 ∈ (𝐺 supp ∅))
28	20, 21, 24, 26, 27	syl22anc 852	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → 𝑐 ∈ (𝐺 supp ∅))
29	28	rabssdv 4022	. . . . . . . 8 ⊢ (𝜑 → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅))
30	29	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅))
31		fsuppimp 9341	. . . . . . . 8 ⊢ (𝐺 finSupp ∅ → (Fun 𝐺 ∧ (𝐺 supp ∅) ∈ Fin))
32		ssfi 9170	. . . . . . . . 9 ⊢ (((𝐺 supp ∅) ∈ Fin ∧ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅)) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin)
33	32	ex 418	. . . . . . . 8 ⊢ ((𝐺 supp ∅) ∈ Fin → ({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin))
34	31, 33	simpl2im 513	. . . . . . 7 ⊢ (𝐺 finSupp ∅ → ({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin))
35	19, 30, 34	sylc 66	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin)
36		fveq2 6879	. . . . . . . . 9 ⊢ (𝑐 = 𝑧 → (𝐹‘𝑐) = (𝐹‘𝑧))
37		fveq2 6879	. . . . . . . . 9 ⊢ (𝑐 = 𝑧 → (𝐺‘𝑐) = (𝐺‘𝑧))
38	36, 37	eleq12d 2854	. . . . . . . 8 ⊢ (𝑐 = 𝑧 → ((𝐹‘𝑐) ∈ (𝐺‘𝑐) ↔ (𝐹‘𝑧) ∈ (𝐺‘𝑧)))
39		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑧 ∈ 𝐵)
40		simprrl 793	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝐹‘𝑧) ∈ (𝐺‘𝑧))
41	38, 39, 40	elrabd 3647	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑧 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)})
42	41	ne0d 4288	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ≠ ∅)
43		ordunifi 9263	. . . . . 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)})
44	15, 35, 42, 43	syl3anc 1398	. . . . 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)})
45	11, 44	eqeltrid 2864	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)})
46	10, 45	sselid 3929	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 ∈ 𝐵)
47		fveq2 6879	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
48		fveq2 6879	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
49	47, 48	eleq12d 2854	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ∈ (𝐺‘𝑥) ↔ (𝐹‘𝑋) ∈ (𝐺‘𝑋)))
50		fveq2 6879	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (𝐹‘𝑐) = (𝐹‘𝑥))
51		fveq2 6879	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (𝐺‘𝑐) = (𝐺‘𝑥))
52	50, 51	eleq12d 2854	. . . . . . 7 ⊢ (𝑐 = 𝑥 → ((𝐹‘𝑐) ∈ (𝐺‘𝑐) ↔ (𝐹‘𝑥) ∈ (𝐺‘𝑥)))
53	52	cbvrabv 3422	. . . . . 6 ⊢ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} = {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) ∈ (𝐺‘𝑥)}
54	49, 53	elrab2 3649	. . . . 5 ⊢ (𝑋 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ↔ (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋)))
55	45, 54	sylib 221	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋)))
56	55	simprd 501	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝐹‘𝑋) ∈ (𝐺‘𝑋))
57		simprrr 794	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))
58	3	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐴 ∈ On)
59	22	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐺:𝐵⟶𝐴)
60	59, 46	ffvelcdmd 7079	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝐺‘𝑋) ∈ 𝐴)
61		onelon 6382	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑋) ∈ 𝐴) → (𝐺‘𝑋) ∈ On)
62	58, 60, 61	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝐺‘𝑋) ∈ On)
63		eloni 6367	. . . . . . . . . 10 ⊢ ((𝐺‘𝑋) ∈ On → Ord (𝐺‘𝑋))
64		ordirr 6375	. . . . . . . . . 10 ⊢ (Ord (𝐺‘𝑋) → ¬ (𝐺‘𝑋) ∈ (𝐺‘𝑋))
65	62, 63, 64	3syl 19	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ¬ (𝐺‘𝑋) ∈ (𝐺‘𝑋))
66		nelneq 2884	. . . . . . . . 9 ⊢ (((𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ¬ (𝐺‘𝑋) ∈ (𝐺‘𝑋)) → ¬ (𝐹‘𝑋) = (𝐺‘𝑋))
67	56, 65, 66	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ¬ (𝐹‘𝑋) = (𝐺‘𝑋))
68		eleq2 2849	. . . . . . . . . 10 ⊢ (𝑤 = 𝑋 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑋))
69		fveq2 6879	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑋 → (𝐹‘𝑤) = (𝐹‘𝑋))
70		fveq2 6879	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑋 → (𝐺‘𝑤) = (𝐺‘𝑋))
71	69, 70	eqeq12d 2776	. . . . . . . . . 10 ⊢ (𝑤 = 𝑋 → ((𝐹‘𝑤) = (𝐺‘𝑤) ↔ (𝐹‘𝑋) = (𝐺‘𝑋)))
72	68, 71	imbi12d 347	. . . . . . . . 9 ⊢ (𝑤 = 𝑋 → ((𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) ↔ (𝑧 ∈ 𝑋 → (𝐹‘𝑋) = (𝐺‘𝑋))))
73	72, 57, 46	rspcdva 3577	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝑧 ∈ 𝑋 → (𝐹‘𝑋) = (𝐺‘𝑋)))
74	67, 73	mtod 201	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ¬ 𝑧 ∈ 𝑋)
75		ssexg 5284	. . . . . . . . . . 11 ⊢ (({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ 𝐵 ∧ 𝐵 ∈ On) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ V)
76	10, 12, 75	sylancr 599	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ V)
77		ssonuni 7780	. . . . . . . . . 10 ⊢ ({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ V → ({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ On → ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ On))
78	76, 15, 77	sylc 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ On)
79	11, 78	eqeltrid 2864	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 ∈ On)
80		onelon 6382	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ On)
81	12, 39, 80	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑧 ∈ On)
82		ontri1 6392	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑧 ∈ On) → (𝑋 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑋))
83	79, 81, 82	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝑋 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑋))
84	74, 83	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 ⊆ 𝑧)
85		elssuni 4899	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} → 𝑧 ⊆ ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)})
86	85, 11	sseqtrrdi 3972	. . . . . . 7 ⊢ (𝑧 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} → 𝑧 ⊆ 𝑋)
87	41, 86	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑧 ⊆ 𝑋)
88	84, 87	eqssd 3948	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 = 𝑧)
89		eleq1 2848	. . . . . . 7 ⊢ (𝑋 = 𝑧 → (𝑋 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤))
90	89	imbi1d 344	. . . . . 6 ⊢ (𝑋 = 𝑧 → ((𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) ↔ (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
91	90	ralbidv 3185	. . . . 5 ⊢ (𝑋 = 𝑧 → (∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) ↔ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
92	88, 91	syl 18	. . . 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))))
93	57, 92	mpbird 260	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))
94	46, 56, 93	3jca 1146	. 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))))
95	9, 94	rexlimddv 3169	1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  ∪ cuni 4867   class class class wbr 5103  {copab 5167  dom cdm 5655  Ord word 6356  Oncon0 6357  Fun wfun 6527   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414   supp csupp 8159  Fincfn 8955   finSupp cfsupp 9334   CNF ccnf 9643

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-supp 8160  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-seqom 8440  df-1o 8458  df-map 8831  df-en 8956  df-fin 8959  df-fsupp 9335  df-cnf 9644

This theorem is used by:  cantnflem1a  9667  cantnflem1b  9668  cantnflem1c  9669  cantnflem1d  9670  cantnflem1  9671