Theorem mpofvex 6204

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 5878	. 2 ⊢ (𝑅𝐹𝑆) = (𝐹‘⟨𝑅, 𝑆⟩)
2		elex 2749	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3	2	alimi 1455	. . . . . . . 8 ⊢ (∀𝑦 𝐶 ∈ 𝑉 → ∀𝑦 𝐶 ∈ V)
4		vex 2741	. . . . . . . . 9 ⊢ 𝑧 ∈ V
5		2ndexg 6169	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (2nd ‘𝑧) ∈ V)
6		nfcv 2319	. . . . . . . . . 10 ⊢ Ⅎ𝑦(2nd ‘𝑧)
7		nfcsb1v 3091	. . . . . . . . . . 11 ⊢ Ⅎ𝑦⦋(2nd ‘𝑧) / 𝑦⦌𝐶
8	7	nfel1 2330	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
9		csbeq1a 3067	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
10	9	eleq1d 2246	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑧) → (𝐶 ∈ V ↔ ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
11	6, 8, 10	spcgf 2820	. . . . . . . . 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 1455	. . . . . 6 ⊢ (∀𝑥∀𝑦 𝐶 ∈ 𝑉 → ∀𝑥⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
15		1stexg 6168	. . . . . . 7 ⊢ (𝑧 ∈ V → (1st ‘𝑧) ∈ V)
16		nfcv 2319	. . . . . . . 8 ⊢ Ⅎ𝑥(1st ‘𝑧)
17		nfcsb1v 3091	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶
18	17	nfel1 2330	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
19		csbeq1a 3067	. . . . . . . . 9 ⊢ (𝑥 = (1st ‘𝑧) → ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
20	19	eleq1d 2246	. . . . . . . 8 ⊢ (𝑥 = (1st ‘𝑧) → (⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V ↔ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
21	16, 18, 20	spcgf 2820	. . . . . . 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 1874	. . . 4 ⊢ (∀𝑥∀𝑦 𝐶 ∈ 𝑉 → ∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
25	24	3ad2ant1 1018	. . 3 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
26		opexg 4229	. . . 4 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
27	26	3adant1 1015	. . 3 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
28		fmpo.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
29		mpomptsx 6198	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
30	28, 29	eqtri 2198	. . . 4 ⊢ 𝐹 = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
31	30	mptfvex 5602	. . 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 2264	1 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝑅𝐹𝑆) ∈ V)

Syntax hints:   → wi 4   ∧ w3a 978  ∀wal 1351   = wceq 1353   ∈ wcel 2148  Vcvv 2738  ⦋csb 3058  {csn 3593  ⟨cop 3596  ∪ ciun 3887   ↦ cmpt 4065   × cxp 4625  ‘cfv 5217  (class class class)co 5875   ∈ cmpo 5877  1^st c1st 6139  2^nd c2nd 6140

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fo 5223  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142

This theorem is referenced by:  mpofvexi  6207  oaexg  6449  omexg  6452  oeiexg  6454