Theorem mpt2fvex 5771

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 5458	. 2 ⊢ (𝑅𝐹𝑆) = (𝐹‘⟨𝑅, 𝑆⟩)
2		elex 2560	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3	2	alimi 1341	. . . . . . . 8 ⊢ (∀y 𝐶 ∈ 𝑉 → ∀y 𝐶 ∈ V)
4		vex 2554	. . . . . . . . 9 ⊢ z ∈ V
5		2ndexg 5737	. . . . . . . . 9 ⊢ (z ∈ V → (2nd ‘z) ∈ V)
6		nfcv 2175	. . . . . . . . . 10 ⊢ Ⅎy(2nd ‘z)
7		nfcsb1v 2876	. . . . . . . . . . 11 ⊢ Ⅎy⦋(2nd ‘z) / y⦌𝐶
8	7	nfel1 2185	. . . . . . . . . 10 ⊢ Ⅎy⦋(2nd ‘z) / y⦌𝐶 ∈ V
9		csbeq1a 2854	. . . . . . . . . . 11 ⊢ (y = (2nd ‘z) → 𝐶 = ⦋(2nd ‘z) / y⦌𝐶)
10	9	eleq1d 2103	. . . . . . . . . 10 ⊢ (y = (2nd ‘z) → (𝐶 ∈ V ↔ ⦋(2nd ‘z) / y⦌𝐶 ∈ V))
11	6, 8, 10	spcgf 2629	. . . . . . . . 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 1341	. . . . . 6 ⊢ (∀x∀y 𝐶 ∈ 𝑉 → ∀x⦋(2nd ‘z) / y⦌𝐶 ∈ V)
15		1stexg 5736	. . . . . . 7 ⊢ (z ∈ V → (1st ‘z) ∈ V)
16		nfcv 2175	. . . . . . . 8 ⊢ Ⅎx(1st ‘z)
17		nfcsb1v 2876	. . . . . . . . 9 ⊢ Ⅎx⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶
18	17	nfel1 2185	. . . . . . . 8 ⊢ Ⅎx⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶 ∈ V
19		csbeq1a 2854	. . . . . . . . 9 ⊢ (x = (1st ‘z) → ⦋(2nd ‘z) / y⦌𝐶 = ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶)
20	19	eleq1d 2103	. . . . . . . 8 ⊢ (x = (1st ‘z) → (⦋(2nd ‘z) / y⦌𝐶 ∈ V ↔ ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶 ∈ V))
21	16, 18, 20	spcgf 2629	. . . . . . 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 1751	. . . 4 ⊢ (∀x∀y 𝐶 ∈ 𝑉 → ∀z⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶 ∈ V)
25	24	3ad2ant1 924	. . 3 ⊢ ((∀x∀y 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ∀z⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶 ∈ V)
26		opexg 3955	. . . 4 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
27	26	3adant1 921	. . 3 ⊢ ((∀x∀y 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
28		fmpt2.1	. . . . 5 ⊢ 𝐹 = (x ∈ A, y ∈ B ↦ 𝐶)
29		mpt2mptsx 5765	. . . . 5 ⊢ (x ∈ A, y ∈ B ↦ 𝐶) = (z ∈ ∪ x ∈ A ({x} × B) ↦ ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶)
30	28, 29	eqtri 2057	. . . 4 ⊢ 𝐹 = (z ∈ ∪ x ∈ A ({x} × B) ↦ ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶)
31	30	mptfvex 5199	. . 3 ⊢ ((∀z⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶 ∈ V ∧ ⟨𝑅, 𝑆⟩ ∈ V) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
32	25, 27, 31	syl2anc 391	. 2 ⊢ ((∀x∀y 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
33	1, 32	syl5eqel 2121	1 ⊢ ((∀x∀y 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝑅𝐹𝑆) ∈ V)

Syntax hints:   → wi 4   ∧ w3a 884  ∀wal 1240   = wceq 1242   ∈ wcel 1390  Vcvv 2551  ⦋csb 2846  {csn 3367  ⟨cop 3370  ∪ ciun 3648   ↦ cmpt 3809   × cxp 4286  ‘cfv 4845  (class class class)co 5455   ↦ cmpt2 5457  1^st c1st 5707  2^nd c2nd 5708

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710

This theorem is referenced by:  mpt2fvexi  5774  oaexg  5967  omexg  5970  oeiexg  5972