Theorem fvmptf 5465

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5449 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 2666	. . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
2		fvmptf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		fvmptf.2	. . . . . 6 ⊢ Ⅎ𝑥𝐶
4	3	nfel1 2264	. . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ V
5		fvmptf.4	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
6		nfmpt1 3979	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵)
7	5, 6	nfcxfr 2250	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
8	7, 2	nffv 5383	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝐴)
9	8, 3	nfeq 2261	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶
10	4, 9	nfim 1532	. . . 4 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)
11		fvmptf.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
12	11	eleq1d 2181	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
13		fveq2 5373	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
14	13, 11	eqeq12d 2127	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝐴) = 𝐶))
15	12, 14	imbi12d 233	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹‘𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)))
16	5	fvmpt2 5456	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵)
17	16	ex 114	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (𝐵 ∈ V → (𝐹‘𝑥) = 𝐵))
18	2, 10, 15, 17	vtoclgaf 2720	. . 3 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶))
19	1, 18	syl5 32	. 2 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ 𝑉 → (𝐹‘𝐴) = 𝐶))
20	19	imp 123	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1312   ∈ wcel 1461  Ⅎwnfc 2240  Vcvv 2655   ↦ cmpt 3947  ‘cfv 5079

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-csb 2970  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087

This theorem is referenced by:  fvmptd3  5466  elfvmptrab1  5467  sumrbdclem  11030  fsum3  11041  isumss  11045