Theorem mpofvex 6218

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

Hypothesis

Ref	Expression
fmpo.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   𝑥,𝐵,𝑦

Allowed substitution hints:   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 5891	. 2 ⊢ (𝑅𝐹𝑆) = (𝐹‘⟨𝑅, 𝑆⟩)
2		elex 2760	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3	2	alimi 1465	. . . . . . . 8 ⊢ (∀𝑦 𝐶 ∈ 𝑉 → ∀𝑦 𝐶 ∈ V)
4		vex 2752	. . . . . . . . 9 ⊢ 𝑧 ∈ V
5		2ndexg 6183	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (2nd ‘𝑧) ∈ V)
6		nfcv 2329	. . . . . . . . . 10 ⊢ Ⅎ𝑦(2nd ‘𝑧)
7		nfcsb1v 3102	. . . . . . . . . . 11 ⊢ Ⅎ𝑦⦋(2nd ‘𝑧) / 𝑦⦌𝐶
8	7	nfel1 2340	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
9		csbeq1a 3078	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
10	9	eleq1d 2256	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑧) → (𝐶 ∈ V ↔ ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
11	6, 8, 10	spcgf 2831	. . . . . . . . 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 1465	. . . . . 6 ⊢ (∀𝑥∀𝑦 𝐶 ∈ 𝑉 → ∀𝑥⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
15		1stexg 6182	. . . . . . 7 ⊢ (𝑧 ∈ V → (1st ‘𝑧) ∈ V)
16		nfcv 2329	. . . . . . . 8 ⊢ Ⅎ𝑥(1st ‘𝑧)
17		nfcsb1v 3102	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶
18	17	nfel1 2340	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
19		csbeq1a 3078	. . . . . . . . 9 ⊢ (𝑥 = (1st ‘𝑧) → ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
20	19	eleq1d 2256	. . . . . . . 8 ⊢ (𝑥 = (1st ‘𝑧) → (⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V ↔ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
21	16, 18, 20	spcgf 2831	. . . . . . 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 1884	. . . 4 ⊢ (∀𝑥∀𝑦 𝐶 ∈ 𝑉 → ∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
25	24	3ad2ant1 1019	. . 3 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
26		opexg 4240	. . . 4 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
27	26	3adant1 1016	. . 3 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
28		fmpo.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
29		mpomptsx 6212	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
30	28, 29	eqtri 2208	. . . 4 ⊢ 𝐹 = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
31	30	mptfvex 5614	. . 3 ⊢ ((∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V ∧ ⟨𝑅, 𝑆⟩ ∈ V) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
32	25, 27, 31	syl2anc 411	. 2 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
33	1, 32	eqeltrid 2274	1 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝑅𝐹𝑆) ∈ V)

Syntax hints:   → wi 4   ∧ w3a 979  ∀wal 1361   = wceq 1363   ∈ wcel 2158  Vcvv 2749  ⦋csb 3069  {csn 3604  ⟨cop 3607  ∪ ciun 3898   ↦ cmpt 4076   × cxp 4636  ‘cfv 5228  (class class class)co 5888   ∈ cmpo 5890  1^st c1st 6153  2^nd c2nd 6154

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fo 5234  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156

This theorem is referenced by:  mpofvexi  6221  oaexg  6463  omexg  6466  oeiexg  6468