Theorem fvmptf 5628

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5612 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 2763	. . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
2		fvmptf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		fvmptf.2	. . . . . 6 ⊢ Ⅎ𝑥𝐶
4	3	nfel1 2343	. . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ V
5		fvmptf.4	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
6		nfmpt1 4111	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵)
7	5, 6	nfcxfr 2329	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
8	7, 2	nffv 5544	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝐴)
9	8, 3	nfeq 2340	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶
10	4, 9	nfim 1583	. . . 4 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)
11		fvmptf.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
12	11	eleq1d 2258	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
13		fveq2 5534	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
14	13, 11	eqeq12d 2204	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝐴) = 𝐶))
15	12, 14	imbi12d 234	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹‘𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)))
16	5	fvmpt2 5619	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵)
17	16	ex 115	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (𝐵 ∈ V → (𝐹‘𝑥) = 𝐵))
18	2, 10, 15, 17	vtoclgaf 2817	. . 3 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶))
19	1, 18	syl5 32	. 2 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ 𝑉 → (𝐹‘𝐴) = 𝐶))
20	19	imp 124	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160  Ⅎwnfc 2319  Vcvv 2752   ↦ cmpt 4079  ‘cfv 5235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243

This theorem is referenced by:  fvmptd3  5629  elfvmptrab1  5630  sumrbdclem  11416  fsum3  11426  isumss  11430  prodrbdclem  11610  prodmodclem2a  11615  zproddc  11618  fprodntrivap  11623  prodssdc  11628  pcmpt  12374