Theorem mpt2fvex 5849

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

Hypothesis

Ref	Expression
fmpt2.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
mpt2fvex	⊢ ((∀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 mpt2fvex

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 5535	. 2 ⊢ (𝑅𝐹𝑆) = (𝐹‘⟨𝑅, 𝑆⟩)
2		elex 2610	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3	2	alimi 1384	. . . . . . . 8 ⊢ (∀𝑦 𝐶 ∈ 𝑉 → ∀𝑦 𝐶 ∈ V)
4		vex 2604	. . . . . . . . 9 ⊢ 𝑧 ∈ V
5		2ndexg 5815	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (2nd ‘𝑧) ∈ V)
6		nfcv 2219	. . . . . . . . . 10 ⊢ Ⅎ𝑦(2nd ‘𝑧)
7		nfcsb1v 2938	. . . . . . . . . . 11 ⊢ Ⅎ𝑦⦋(2nd ‘𝑧) / 𝑦⦌𝐶
8	7	nfel1 2229	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
9		csbeq1a 2916	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
10	9	eleq1d 2147	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑧) → (𝐶 ∈ V ↔ ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
11	6, 8, 10	spcgf 2680	. . . . . . . . 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 1384	. . . . . 6 ⊢ (∀𝑥∀𝑦 𝐶 ∈ 𝑉 → ∀𝑥⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
15		1stexg 5814	. . . . . . 7 ⊢ (𝑧 ∈ V → (1st ‘𝑧) ∈ V)
16		nfcv 2219	. . . . . . . 8 ⊢ Ⅎ𝑥(1st ‘𝑧)
17		nfcsb1v 2938	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶
18	17	nfel1 2229	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V
19		csbeq1a 2916	. . . . . . . . 9 ⊢ (𝑥 = (1st ‘𝑧) → ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
20	19	eleq1d 2147	. . . . . . . 8 ⊢ (𝑥 = (1st ‘𝑧) → (⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V ↔ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V))
21	16, 18, 20	spcgf 2680	. . . . . . 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 1795	. . . 4 ⊢ (∀𝑥∀𝑦 𝐶 ∈ 𝑉 → ∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
25	24	3ad2ant1 959	. . 3 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V)
26		opexg 3983	. . . 4 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
27	26	3adant1 956	. . 3 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
28		fmpt2.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
29		mpt2mptsx 5843	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
30	28, 29	eqtri 2101	. . . 4 ⊢ 𝐹 = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
31	30	mptfvex 5277	. . 3 ⊢ ((∀𝑧⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ∈ V ∧ ⟨𝑅, 𝑆⟩ ∈ V) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
32	25, 27, 31	syl2anc 403	. 2 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
33	1, 32	syl5eqel 2165	1 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝑅𝐹𝑆) ∈ V)

Syntax hints:   → wi 4   ∧ w3a 919  ∀wal 1282   = wceq 1284   ∈ wcel 1433  Vcvv 2601  ⦋csb 2908  {csn 3398  ⟨cop 3401  ∪ ciun 3678   ↦ cmpt 3839   × cxp 4361  ‘cfv 4922  (class class class)co 5532   ↦ cmpt2 5534  1^st c1st 5785  2^nd c2nd 5786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788

This theorem is referenced by:  mpt2fvexi  5852  oaexg  6051  omexg  6054  oeiexg  6056