Theorem mpofvex 6109

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 5785	. 2 ⊢ (𝑅𝐹𝑆) = (𝐹‘⟨𝑅, 𝑆⟩)
2		elex 2700	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3	2	alimi 1432	. . . . . . . 8 ⊢ (∀𝑦 𝐶 ∈ 𝑉 → ∀𝑦 𝐶 ∈ V)
4		vex 2692	. . . . . . . . 9 ⊢ 𝑧 ∈ V
5		2ndexg 6074	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (2nd ‘𝑧) ∈ V)
6		nfcv 2282	. . . . . . . . . 10 ⊢ Ⅎ𝑦(2nd ‘𝑧)
7		nfcsb1v 3040	. . . . . . . . . . 11 ⊢ Ⅎ𝑦⦋(2nd ‘𝑧) / 𝑦⦌𝐶
8	7	nfel1 2293	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
9		csbeq1a 3016	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
10	9	eleq1d 2209	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑧) → (𝐶 ∈ V ↔ ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
11	6, 8, 10	spcgf 2771	. . . . . . . . 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 1432	. . . . . 6 ⊢ (∀𝑥∀𝑦 𝐶 ∈ 𝑉 → ∀𝑥⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
15		1stexg 6073	. . . . . . 7 ⊢ (𝑧 ∈ V → (1st ‘𝑧) ∈ V)
16		nfcv 2282	. . . . . . . 8 ⊢ Ⅎ𝑥(1st ‘𝑧)
17		nfcsb1v 3040	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶
18	17	nfel1 2293	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
19		csbeq1a 3016	. . . . . . . . 9 ⊢ (𝑥 = (1st ‘𝑧) → ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
20	19	eleq1d 2209	. . . . . . . 8 ⊢ (𝑥 = (1st ‘𝑧) → (⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V ↔ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
21	16, 18, 20	spcgf 2771	. . . . . . 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 1847	. . . 4 ⊢ (∀𝑥∀𝑦 𝐶 ∈ 𝑉 → ∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
25	24	3ad2ant1 1003	. . 3 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
26		opexg 4158	. . . 4 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
27	26	3adant1 1000	. . 3 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
28		fmpo.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
29		mpomptsx 6103	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
30	28, 29	eqtri 2161	. . . 4 ⊢ 𝐹 = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
31	30	mptfvex 5514	. . 3 ⊢ ((∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V ∧ ⟨𝑅, 𝑆⟩ ∈ V) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
32	25, 27, 31	syl2anc 409	. 2 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
33	1, 32	eqeltrid 2227	1 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝑅𝐹𝑆) ∈ V)

Syntax hints:   → wi 4   ∧ w3a 963  ∀wal 1330   = wceq 1332   ∈ wcel 1481  Vcvv 2689  ⦋csb 3007  {csn 3532  ⟨cop 3535  ∪ ciun 3821   ↦ cmpt 3997   × cxp 4545  ‘cfv 5131  (class class class)co 5782   ∈ cmpo 5784  1^st c1st 6044  2^nd c2nd 6045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fo 5137  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047

This theorem is referenced by:  mpofvexi  6112  oaexg  6352  omexg  6355  oeiexg  6357