Theorem finixpnum 37634

Description: A finite Cartesian product of numerable sets is numerable. (Contributed by Brendan Leahy, 24-Feb-2019.)

Assertion

Ref	Expression
finixpnum	⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ dom card) → X𝑥 ∈ 𝐴 𝐵 ∈ dom card)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem finixpnum

Dummy variables 𝑣 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 3306	. . . 4 ⊢ (𝑤 = ∅ → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ ∅ 𝐵 ∈ dom card))
2		ixpeq1 8927	. . . . . 6 ⊢ (𝑤 = ∅ → X𝑥 ∈ 𝑤 𝐵 = X𝑥 ∈ ∅ 𝐵)
3		ixp0x 8945	. . . . . 6 ⊢ X𝑥 ∈ ∅ 𝐵 = {∅}
4	2, 3	eqtrdi 2787	. . . . 5 ⊢ (𝑤 = ∅ → X𝑥 ∈ 𝑤 𝐵 = {∅})
5	4	eleq1d 2820	. . . 4 ⊢ (𝑤 = ∅ → (X𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ {∅} ∈ dom card))
6	1, 5	imbi12d 344	. . 3 ⊢ (𝑤 = ∅ → ((∀𝑥 ∈ 𝑤 𝐵 ∈ dom card → X𝑥 ∈ 𝑤 𝐵 ∈ dom card) ↔ (∀𝑥 ∈ ∅ 𝐵 ∈ dom card → {∅} ∈ dom card)))
7		raleq 3306	. . . 4 ⊢ (𝑤 = 𝑦 → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ 𝑦 𝐵 ∈ dom card))
8		ixpeq1 8927	. . . . 5 ⊢ (𝑤 = 𝑦 → X𝑥 ∈ 𝑤 𝐵 = X𝑥 ∈ 𝑦 𝐵)
9	8	eleq1d 2820	. . . 4 ⊢ (𝑤 = 𝑦 → (X𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ X𝑥 ∈ 𝑦 𝐵 ∈ dom card))
10	7, 9	imbi12d 344	. . 3 ⊢ (𝑤 = 𝑦 → ((∀𝑥 ∈ 𝑤 𝐵 ∈ dom card → X𝑥 ∈ 𝑤 𝐵 ∈ dom card) ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card)))
11		raleq 3306	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
12		ralunb 4177	. . . . . 6 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ∀𝑥 ∈ {𝑧}𝐵 ∈ dom card))
13		vex 3468	. . . . . . . 8 ⊢ 𝑧 ∈ V
14		ralsnsg 4651	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝐵 ∈ dom card ↔ [𝑧 / 𝑥]𝐵 ∈ dom card))
15		sbcel1g 4396	. . . . . . . . 9 ⊢ (𝑧 ∈ V → ([𝑧 / 𝑥]𝐵 ∈ dom card ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card))
16	14, 15	bitrd 279	. . . . . . . 8 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝐵 ∈ dom card ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card))
17	13, 16	ax-mp 5	. . . . . . 7 ⊢ (∀𝑥 ∈ {𝑧}𝐵 ∈ dom card ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card)
18	17	anbi2i 623	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ∀𝑥 ∈ {𝑧}𝐵 ∈ dom card) ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card))
19	12, 18	bitri 275	. . . . 5 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card))
20	11, 19	bitrdi 287	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card)))
21		ixpeq1 8927	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → X𝑥 ∈ 𝑤 𝐵 = X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
22	21	eleq1d 2820	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (X𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
23	20, 22	imbi12d 344	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((∀𝑥 ∈ 𝑤 𝐵 ∈ dom card → X𝑥 ∈ 𝑤 𝐵 ∈ dom card) ↔ ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
24		raleq 3306	. . . 4 ⊢ (𝑤 = 𝐴 → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ dom card))
25		ixpeq1 8927	. . . . 5 ⊢ (𝑤 = 𝐴 → X𝑥 ∈ 𝑤 𝐵 = X𝑥 ∈ 𝐴 𝐵)
26	25	eleq1d 2820	. . . 4 ⊢ (𝑤 = 𝐴 → (X𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ X𝑥 ∈ 𝐴 𝐵 ∈ dom card))
27	24, 26	imbi12d 344	. . 3 ⊢ (𝑤 = 𝐴 → ((∀𝑥 ∈ 𝑤 𝐵 ∈ dom card → X𝑥 ∈ 𝑤 𝐵 ∈ dom card) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ dom card → X𝑥 ∈ 𝐴 𝐵 ∈ dom card)))
28		snfi 9062	. . . 4 ⊢ {∅} ∈ Fin
29		finnum 9967	. . . 4 ⊢ ({∅} ∈ Fin → {∅} ∈ dom card)
30	28, 29	mp1i 13	. . 3 ⊢ (∀𝑥 ∈ ∅ 𝐵 ∈ dom card → {∅} ∈ dom card)
31		pm2.27 42	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → X𝑥 ∈ 𝑦 𝐵 ∈ dom card))
32		xpnum 9970	. . . . . . . . . . 11 ⊢ ((X𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ∈ dom card)
33	32	ancoms 458	. . . . . . . . . 10 ⊢ ((⦋𝑧 / 𝑥⦌𝐵 ∈ dom card ∧ X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ∈ dom card)
34		xp1st 8025	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → (1st ‘𝑤) ∈ X𝑥 ∈ 𝑦 𝐵)
35		ixpfn 8922	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑤) ∈ X𝑥 ∈ 𝑦 𝐵 → (1st ‘𝑤) Fn 𝑦)
36	34, 35	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → (1st ‘𝑤) Fn 𝑦)
37		fvex 6894	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘𝑤) ∈ V
38	13, 37	fnsn 6599	. . . . . . . . . . . . . . 15 ⊢ {⟨𝑧, (2nd ‘𝑤)⟩} Fn {𝑧}
39	36, 38	jctir 520	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → ((1st ‘𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd ‘𝑤)⟩} Fn {𝑧}))
40		disjsn 4692	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
41	40	biimpri 228	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑦 ∩ {𝑧}) = ∅)
42		fnun 6657	. . . . . . . . . . . . . 14 ⊢ ((((1st ‘𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd ‘𝑤)⟩} Fn {𝑧}) ∧ (𝑦 ∩ {𝑧}) = ∅) → ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) Fn (𝑦 ∪ {𝑧}))
43	39, 41, 42	syl2anr 597	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) Fn (𝑦 ∪ {𝑧}))
44		fvex 6894	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘𝑤) ∈ V
45	44	elixp 8923	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑤) ∈ X𝑥 ∈ 𝑦 𝐵 ↔ ((1st ‘𝑤) Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 ((1st ‘𝑤)‘𝑥) ∈ 𝐵))
46	34, 45	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → ((1st ‘𝑤) Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 ((1st ‘𝑤)‘𝑥) ∈ 𝐵))
47		fvun1 6975	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd ‘𝑤)⟩} Fn {𝑧} ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑥 ∈ 𝑦)) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = ((1st ‘𝑤)‘𝑥))
48	38, 47	mp3an2 1451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝑤) Fn 𝑦 ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑥 ∈ 𝑦)) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = ((1st ‘𝑤)‘𝑥))
49	48	anassrs 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((1st ‘𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥 ∈ 𝑦) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = ((1st ‘𝑤)‘𝑥))
50	49	eleq1d 2820	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1st ‘𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥 ∈ 𝑦) → ((((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ ((1st ‘𝑤)‘𝑥) ∈ 𝐵))
51	50	biimprd 248	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1st ‘𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥 ∈ 𝑦) → (((1st ‘𝑤)‘𝑥) ∈ 𝐵 → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵))
52	51	ralimdva 3153	. . . . . . . . . . . . . . . . 17 ⊢ (((1st ‘𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) → (∀𝑥 ∈ 𝑦 ((1st ‘𝑤)‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵))
53	52	ancoms 458	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∩ {𝑧}) = ∅ ∧ (1st ‘𝑤) Fn 𝑦) → (∀𝑥 ∈ 𝑦 ((1st ‘𝑤)‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵))
54	53	impr 454	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∩ {𝑧}) = ∅ ∧ ((1st ‘𝑤) Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 ((1st ‘𝑤)‘𝑥) ∈ 𝐵)) → ∀𝑥 ∈ 𝑦 (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
55	41, 46, 54	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ∀𝑥 ∈ 𝑦 (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
56		vsnid 4644	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑧 ∈ {𝑧}
57	41, 56	jctir 520	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧}))
58		fvun2 6976	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd ‘𝑤)⟩} Fn {𝑧} ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧})) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧))
59	38, 58	mp3an2 1451	. . . . . . . . . . . . . . . . . 18 ⊢ (((1st ‘𝑤) Fn 𝑦 ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧})) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧))
60	36, 57, 59	syl2anr 597	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧))
61		csbfv 6931	. . . . . . . . . . . . . . . . 17 ⊢ ⦋𝑧 / 𝑥⦌(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑧)
62	13, 37	fvsn 7178	. . . . . . . . . . . . . . . . . 18 ⊢ ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧) = (2nd ‘𝑤)
63	62	eqcomi 2745	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘𝑤) = ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧)
64	60, 61, 63	3eqtr4g 2796	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ⦋𝑧 / 𝑥⦌(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = (2nd ‘𝑤))
65		xp2nd 8026	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → (2nd ‘𝑤) ∈ ⦋𝑧 / 𝑥⦌𝐵)
66	65	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → (2nd ‘𝑤) ∈ ⦋𝑧 / 𝑥⦌𝐵)
67	64, 66	eqeltrd 2835	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ⦋𝑧 / 𝑥⦌(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ ⦋𝑧 / 𝑥⦌𝐵)
68		ralsnsg 4651	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧} (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ [𝑧 / 𝑥](((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵))
69	13, 68	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ {𝑧} (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ [𝑧 / 𝑥](((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
70		sbcel12 4391	. . . . . . . . . . . . . . . 16 ⊢ ([𝑧 / 𝑥](((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ ⦋𝑧 / 𝑥⦌𝐵)
71	69, 70	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ {𝑧} (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ ⦋𝑧 / 𝑥⦌𝐵)
72	67, 71	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ∀𝑥 ∈ {𝑧} (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
73		ralun 4178	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝑦 (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ {𝑧} (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝑦 ∪ {𝑧})(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
74	55, 72, 73	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ∀𝑥 ∈ (𝑦 ∪ {𝑧})(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
75		snex 5411	. . . . . . . . . . . . . . 15 ⊢ {⟨𝑧, (2nd ‘𝑤)⟩} ∈ V
76	44, 75	unex 7743	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) ∈ V
77	76	elixp 8923	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵))
78	43, 74, 77	sylanbrc 583	. . . . . . . . . . . 12 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
79	78	fmpttd 7110	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})):(X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)⟶X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
80		ixpfn 8922	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → 𝑢 Fn (𝑦 ∪ {𝑧}))
81		ssun1 4158	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
82		fnssres 6666	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ 𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑢 ↾ 𝑦) Fn 𝑦)
83	80, 81, 82	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢 ↾ 𝑦) Fn 𝑦)
84		vex 3468	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑢 ∈ V
85	84	elixp 8923	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ (𝑢 Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢‘𝑥) ∈ 𝐵))
86		ssralv 4032	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 (𝑢‘𝑥) ∈ 𝐵))
87	81, 86	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 (𝑢‘𝑥) ∈ 𝐵)
88		fvres 6900	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝑦 → ((𝑢 ↾ 𝑦)‘𝑥) = (𝑢‘𝑥))
89	88	eleq1d 2820	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝑦 → (((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵 ↔ (𝑢‘𝑥) ∈ 𝐵))
90	89	biimprd 248	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝑦 → ((𝑢‘𝑥) ∈ 𝐵 → ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵))
91	90	ralimia 3071	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑥 ∈ 𝑦 (𝑢‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵)
92	87, 91	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵)
93	92	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵)
94	85, 93	sylbi 217	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵)
95	84	resex 6021	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ↾ 𝑦) ∈ V
96	95	elixp 8923	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ↾ 𝑦) ∈ X𝑥 ∈ 𝑦 𝐵 ↔ ((𝑢 ↾ 𝑦) Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵))
97	83, 94, 96	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢 ↾ 𝑦) ∈ X𝑥 ∈ 𝑦 𝐵)
98		ssun2 4159	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
99	98, 56	sselii 3960	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
100		csbeq1 3882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
101	100	fvixp 8921	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ X𝑤 ∈ (𝑦 ∪ {𝑧})⦋𝑤 / 𝑥⦌𝐵 ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (𝑢‘𝑧) ∈ ⦋𝑧 / 𝑥⦌𝐵)
102	99, 101	mpan2 691	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ X𝑤 ∈ (𝑦 ∪ {𝑧})⦋𝑤 / 𝑥⦌𝐵 → (𝑢‘𝑧) ∈ ⦋𝑧 / 𝑥⦌𝐵)
103		nfcv 2899	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑤𝐵
104		nfcsb1v 3903	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵
105		csbeq1a 3893	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵)
106	103, 104, 105	cbvixp 8933	. . . . . . . . . . . . . . . 16 ⊢ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = X𝑤 ∈ (𝑦 ∪ {𝑧})⦋𝑤 / 𝑥⦌𝐵
107	102, 106	eleq2s 2853	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢‘𝑧) ∈ ⦋𝑧 / 𝑥⦌𝐵)
108		opelxpi 5696	. . . . . . . . . . . . . . 15 ⊢ (((𝑢 ↾ 𝑦) ∈ X𝑥 ∈ 𝑦 𝐵 ∧ (𝑢‘𝑧) ∈ ⦋𝑧 / 𝑥⦌𝐵) → ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵))
109	97, 107, 108	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵))
110	109	adantl 481	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵))
111		disj3 4434	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ 𝑦 = (𝑦 ∖ {𝑧}))
112	40, 111	sylbb1 237	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑧 ∈ 𝑦 → 𝑦 = (𝑦 ∖ {𝑧}))
113		difun2 4461	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = (𝑦 ∖ {𝑧})
114	112, 113	eqtr4di 2789	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑧 ∈ 𝑦 → 𝑦 = ((𝑦 ∪ {𝑧}) ∖ {𝑧}))
115	114	reseq2d 5971	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑢 ↾ 𝑦) = (𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})))
116	115	uneq1d 4147	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}) = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
117	116	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}) = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
118		fvex 6894	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢‘𝑧) ∈ V
119	95, 118	op1std 8003	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → (1st ‘𝑤) = (𝑢 ↾ 𝑦))
120	95, 118	op2ndd 8004	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → (2nd ‘𝑤) = (𝑢‘𝑧))
121	120	opeq2d 4861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → ⟨𝑧, (2nd ‘𝑤)⟩ = ⟨𝑧, (𝑢‘𝑧)⟩)
122	121	sneqd 4618	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → {⟨𝑧, (2nd ‘𝑤)⟩} = {⟨𝑧, (𝑢‘𝑧)⟩})
123	119, 122	uneq12d 4149	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) = ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
124		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})) = (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))
125		snex 5411	. . . . . . . . . . . . . . . . . 18 ⊢ {⟨𝑧, (𝑢‘𝑧)⟩} ∈ V
126	95, 125	unex 7743	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}) ∈ V
127	123, 124, 126	fvmpt 6991	. . . . . . . . . . . . . . . 16 ⊢ (⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩) = ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
128	109, 127	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩) = ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
129	128	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩) = ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
130		fnsnsplit 7181	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
131	80, 99, 130	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
132	131	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
133	117, 129, 132	3eqtr4rd 2782	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → 𝑢 = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩))
134		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘𝑣) = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩))
135	134	rspceeqv 3629	. . . . . . . . . . . . 13 ⊢ ((⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑢 = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩)) → ∃𝑣 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)𝑢 = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘𝑣))
136	110, 133, 135	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ∃𝑣 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)𝑢 = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘𝑣))
137	136	ralrimiva 3133	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ 𝑦 → ∀𝑢 ∈ X 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵∃𝑣 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)𝑢 = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘𝑣))
138		dffo3 7097	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})):(X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)–onto→X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})):(X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)⟶X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∧ ∀𝑢 ∈ X 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵∃𝑣 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)𝑢 = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘𝑣)))
139	79, 137, 138	sylanbrc 583	. . . . . . . . . 10 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})):(X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)–onto→X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
140		fonum 10077	. . . . . . . . . 10 ⊢ (((X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ∈ dom card ∧ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})):(X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)–onto→X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)
141	33, 139, 140	syl2anr 597	. . . . . . . . 9 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ (⦋𝑧 / 𝑥⦌𝐵 ∈ dom card ∧ X𝑥 ∈ 𝑦 𝐵 ∈ dom card)) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)
142	141	expr 456	. . . . . . . 8 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → (X𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
143	31, 142	syl9r 78	. . . . . . 7 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
144	143	expimpd 453	. . . . . 6 ⊢ (¬ 𝑧 ∈ 𝑦 → ((⦋𝑧 / 𝑥⦌𝐵 ∈ dom card ∧ ∀𝑥 ∈ 𝑦 𝐵 ∈ dom card) → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
145	144	ancomsd 465	. . . . 5 ⊢ (¬ 𝑧 ∈ 𝑦 → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
146	145	com23 86	. . . 4 ⊢ (¬ 𝑧 ∈ 𝑦 → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
147	146	adantl 481	. . 3 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
148	6, 10, 23, 27, 30, 147	findcard2s 9184	. 2 ⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 𝐵 ∈ dom card → X𝑥 ∈ 𝐴 𝐵 ∈ dom card))
149	148	imp 406	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ dom card) → X𝑥 ∈ 𝐴 𝐵 ∈ dom card)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061  Vcvv 3464  [wsbc 3770  ⦋csb 3879   ∖ cdif 3928   ∪ cun 3929   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  {csn 4606  ⟨cop 4612   ↦ cmpt 5206   × cxp 5657  dom cdm 5659   ↾ cres 5661   Fn wfn 6531  ⟶wf 6532  –onto→wfo 6534  ‘cfv 6536  1^st c1st 7991  2^nd c2nd 7992  Xcixp 8916  Fincfn 8964  cardccrd 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-omul 8490  df-er 8724  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-fin 8968  df-card 9958  df-acn 9961

This theorem is referenced by:  poimirlem32  37681