Theorem fvmptf 5188

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5173 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
fvmptf.1	⊢ ℲxA
fvmptf.2	⊢ Ⅎx𝐶
fvmptf.3	⊢ (x = A → B = 𝐶)
fvmptf.4	⊢ 𝐹 = (x ∈ 𝐷 ↦ B)

Assertion

Ref	Expression
fvmptf	⊢ ((A ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘A) = 𝐶)

Distinct variable group:   x,𝐷

Allowed substitution hints:   A(x)   B(x)   𝐶(x)   𝐹(x)   𝑉(x)

Proof of Theorem fvmptf

Step	Hyp	Ref	Expression
1		elex 2543	. . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
2		fvmptf.1	. . . 4 ⊢ ℲxA
3		fvmptf.2	. . . . . 6 ⊢ Ⅎx𝐶
4	3	nfel1 2170	. . . . 5 ⊢ Ⅎx 𝐶 ∈ V
5		fvmptf.4	. . . . . . . 8 ⊢ 𝐹 = (x ∈ 𝐷 ↦ B)
6		nfmpt1 3824	. . . . . . . 8 ⊢ Ⅎx(x ∈ 𝐷 ↦ B)
7	5, 6	nfcxfr 2157	. . . . . . 7 ⊢ Ⅎx𝐹
8	7, 2	nffv 5110	. . . . . 6 ⊢ Ⅎx(𝐹‘A)
9	8, 3	nfeq 2167	. . . . 5 ⊢ Ⅎx(𝐹‘A) = 𝐶
10	4, 9	nfim 1446	. . . 4 ⊢ Ⅎx(𝐶 ∈ V → (𝐹‘A) = 𝐶)
11		fvmptf.3	. . . . . 6 ⊢ (x = A → B = 𝐶)
12	11	eleq1d 2088	. . . . 5 ⊢ (x = A → (B ∈ V ↔ 𝐶 ∈ V))
13		fveq2 5103	. . . . . 6 ⊢ (x = A → (𝐹‘x) = (𝐹‘A))
14	13, 11	eqeq12d 2036	. . . . 5 ⊢ (x = A → ((𝐹‘x) = B ↔ (𝐹‘A) = 𝐶))
15	12, 14	imbi12d 223	. . . 4 ⊢ (x = A → ((B ∈ V → (𝐹‘x) = B) ↔ (𝐶 ∈ V → (𝐹‘A) = 𝐶)))
16	5	fvmpt2 5179	. . . . 5 ⊢ ((x ∈ 𝐷 ∧ B ∈ V) → (𝐹‘x) = B)
17	16	ex 108	. . . 4 ⊢ (x ∈ 𝐷 → (B ∈ V → (𝐹‘x) = B))
18	2, 10, 15, 17	vtoclgaf 2595	. . 3 ⊢ (A ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘A) = 𝐶))
19	1, 18	syl5 28	. 2 ⊢ (A ∈ 𝐷 → (𝐶 ∈ 𝑉 → (𝐹‘A) = 𝐶))
20	19	imp 115	1 ⊢ ((A ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘A) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374  Ⅎwnfc 2147  Vcvv 2535   ↦ cmpt 3792  ‘cfv 4829

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-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

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-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-iota 4794  df-fun 4831  df-fv 4837

This theorem is referenced by: (None)