Theorem mpt2fvex 5752

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 5439	. 2 ⊢ (𝑅𝐹𝑆) = (𝐹‘⟨𝑅, 𝑆⟩)
2		elex 2543	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3	2	alimi 1324	. . . . . . . 8 ⊢ (∀y 𝐶 ∈ 𝑉 → ∀y 𝐶 ∈ V)
4		vex 2538	. . . . . . . . 9 ⊢ z ∈ V
5		2ndexg 5718	. . . . . . . . 9 ⊢ (z ∈ V → (2nd ‘z) ∈ V)
6		nfcv 2160	. . . . . . . . . 10 ⊢ Ⅎy(2nd ‘z)
7		nfcsb1v 2859	. . . . . . . . . . 11 ⊢ Ⅎy⦋(2nd ‘z) / y⦌𝐶
8	7	nfel1 2170	. . . . . . . . . 10 ⊢ Ⅎy⦋(2nd ‘z) / y⦌𝐶 ∈ V
9		csbeq1a 2837	. . . . . . . . . . 11 ⊢ (y = (2nd ‘z) → 𝐶 = ⦋(2nd ‘z) / y⦌𝐶)
10	9	eleq1d 2088	. . . . . . . . . 10 ⊢ (y = (2nd ‘z) → (𝐶 ∈ V ↔ ⦋(2nd ‘z) / y⦌𝐶 ∈ V))
11	6, 8, 10	spcgf 2612	. . . . . . . . 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 1324	. . . . . 6 ⊢ (∀x∀y 𝐶 ∈ 𝑉 → ∀x⦋(2nd ‘z) / y⦌𝐶 ∈ V)
15		1stexg 5717	. . . . . . 7 ⊢ (z ∈ V → (1st ‘z) ∈ V)
16		nfcv 2160	. . . . . . . 8 ⊢ Ⅎx(1st ‘z)
17		nfcsb1v 2859	. . . . . . . . 9 ⊢ Ⅎx⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶
18	17	nfel1 2170	. . . . . . . 8 ⊢ Ⅎx⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶 ∈ V
19		csbeq1a 2837	. . . . . . . . 9 ⊢ (x = (1st ‘z) → ⦋(2nd ‘z) / y⦌𝐶 = ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶)
20	19	eleq1d 2088	. . . . . . . 8 ⊢ (x = (1st ‘z) → (⦋(2nd ‘z) / y⦌𝐶 ∈ V ↔ ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶 ∈ V))
21	16, 18, 20	spcgf 2612	. . . . . . 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 1736	. . . 4 ⊢ (∀x∀y 𝐶 ∈ 𝑉 → ∀z⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶 ∈ V)
25	24	3ad2ant1 913	. . 3 ⊢ ((∀x∀y 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ∀z⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶 ∈ V)
26		opexg 3938	. . . 4 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
27	26	3adant1 910	. . 3 ⊢ ((∀x∀y 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → ⟨𝑅, 𝑆⟩ ∈ V)
28		fmpt2.1	. . . . 5 ⊢ 𝐹 = (x ∈ A, y ∈ B ↦ 𝐶)
29		mpt2mptsx 5746	. . . . 5 ⊢ (x ∈ A, y ∈ B ↦ 𝐶) = (z ∈ ∪ x ∈ A ({x} × B) ↦ ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶)
30	28, 29	eqtri 2042	. . . 4 ⊢ 𝐹 = (z ∈ ∪ x ∈ A ({x} × B) ↦ ⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶)
31	30	mptfvex 5181	. . 3 ⊢ ((∀z⦋(1st ‘z) / x⦌⦋(2nd ‘z) / y⦌𝐶 ∈ V ∧ ⟨𝑅, 𝑆⟩ ∈ V) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
32	25, 27, 31	syl2anc 393	. 2 ⊢ ((∀x∀y 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝐹‘⟨𝑅, 𝑆⟩) ∈ V)
33	1, 32	syl5eqel 2106	1 ⊢ ((∀x∀y 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝑅𝐹𝑆) ∈ V)

Syntax hints:   → wi 4   ∧ w3a 873  ∀wal 1226   = wceq 1228   ∈ wcel 1374  Vcvv 2535  ⦋csb 2829  {csn 3350  ⟨cop 3353  ∪ ciun 3631   ↦ cmpt 3792   × cxp 4270  ‘cfv 4829  (class class class)co 5436   ↦ cmpt2 5438  1^st c1st 5688  2^nd c2nd 5689

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691

This theorem is referenced by:  mpt2fvexi  5755  oaexg  5943  omexg  5946  oeiexg  5948