Theorem mpofvex 6435

Description: Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Hypothesis

Ref	Expression
mpofvex.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
mpofvex	⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝑅𝐹𝑆) ∈ V)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem mpofvex

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 6082	. 2 ⊢ (𝑅𝐹𝑆) = (𝐹‘⟨𝑅, 𝑆⟩)
2		elex 2833	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3	2	alimi 1508	. . . . . . . 8 ⊢ (∀𝑦 𝐶 ∈ 𝑉 → ∀𝑦 𝐶 ∈ V)
4		vex 2824	. . . . . . . . 9 ⊢ 𝑧 ∈ V
5		2ndexg 6396	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (2nd ‘𝑧) ∈ V)
6		nfcv 2392	. . . . . . . . . 10 ⊢ Ⅎ𝑦(2nd ‘𝑧)
7		nfcsb1v 3180	. . . . . . . . . . 11 ⊢ Ⅎ𝑦⦋(2nd ‘𝑧) / 𝑦⦌𝐶
8	7	nfel1 2403	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
9		csbeq1a 3156	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
10	9	eleq1d 2307	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑧) → (𝐶 ∈ V ↔ ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
11	6, 8, 10	spcgf 2907	. . . . . . . . 9 ⊢ ((2nd ‘𝑧) ∈ V → (∀𝑦 𝐶 ∈ V → ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
12	4, 5, 11	mp2b 8	. . . . . . . 8 ⊢ (∀𝑦 𝐶 ∈ V → ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
13	3, 12	syl 14	. . . . . . 7 ⊢ (∀𝑦 𝐶 ∈ 𝑉 → ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
14	13	alimi 1508	. . . . . 6 ⊢ (∀𝑥∀𝑦 𝐶 ∈ 𝑉 → ∀𝑥⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
15		1stexg 6395	. . . . . . 7 ⊢ (𝑧 ∈ V → (1st ‘𝑧) ∈ V)
16		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑥(1st ‘𝑧)
17		nfcsb1v 3180	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶
18	17	nfel1 2403	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
19		csbeq1a 3156	. . . . . . . . 9 ⊢ (𝑥 = (1st ‘𝑧) → ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
20	19	eleq1d 2307	. . . . . . . 8 ⊢ (𝑥 = (1st ‘𝑧) → (⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V ↔ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
21	16, 18, 20	spcgf 2907	. . . . . . 7 ⊢ ((1st ‘𝑧) ∈ V → (∀𝑥⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V → ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
22	4, 15, 21	mp2b 8	. . . . . 6 ⊢ (∀𝑥⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V → ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
23	14, 22	syl 14	. . . . 5 ⊢ (∀𝑥∀𝑦 𝐶 ∈ 𝑉 → ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
24	23	alrimiv 1927	. . . 4 ⊢ (∀𝑥∀𝑦 𝐶 ∈ 𝑉 → ∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
25	24	3ad2ant1 1049	. . 3 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
26		opexg 4366	. . . 4 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
27	26	3adant1 1046	. . 3 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
28		mpofvex.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
29		mpomptsx 6427	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
30	28, 29	eqtri 2259	. . . 4 ⊢ 𝐹 = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
31	30	mptfvex 5788	. . 3 ⊢ ((∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V ∧ ⟨𝑅, 𝑆⟩ ∈ V) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
32	25, 27, 31	syl2anc 415	. 2 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
33	1, 32	eqeltrid 2325	1 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝑅𝐹𝑆) ∈ V)

Syntax hints:   → wi 4   ∧ w3a 1009  ∀wal 1400   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ⦋csb 3147  {csn 3708  ⟨cop 3711  ∪ ciun 4010   ↦ cmpt 4190   × cxp 4770  ‘cfv 5375  (class class class)co 6079   ∈ cmpo 6081  1^st c1st 6366  2^nd c2nd 6367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fo 5381  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369

This theorem is referenced by:  mpofvexi  6436  oaexg  6715  omexg  6718  oeiexg  6720  rhmex  14447  clwwlknon  16653