Theorem fvmptf 5695

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5678 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 2788	. . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
2		fvmptf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		fvmptf.2	. . . . . 6 ⊢ Ⅎ𝑥𝐶
4	3	nfel1 2361	. . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ V
5		fvmptf.4	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
6		nfmpt1 4153	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵)
7	5, 6	nfcxfr 2347	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
8	7, 2	nffv 5609	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝐴)
9	8, 3	nfeq 2358	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶
10	4, 9	nfim 1596	. . . 4 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)
11		fvmptf.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
12	11	eleq1d 2276	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
13		fveq2 5599	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
14	13, 11	eqeq12d 2222	. . . . 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 5686	. . . . 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 2843	. . 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 1373   ∈ wcel 2178  Ⅎwnfc 2337  Vcvv 2776   ↦ cmpt 4121  ‘cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298

This theorem is referenced by:  fvmptd3  5696  elfvmptrab1  5697  sumrbdclem  11803  fsum3  11813  isumss  11817  prodrbdclem  11997  prodmodclem2a  12002  zproddc  12005  fprodntrivap  12010  prodssdc  12015  pcmpt  12781