Theorem fvmpt3 5643

Description: Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Hypotheses

Ref	Expression
fvmpt3.a	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
fvmpt3.b	⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
fvmpt3.c	⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
fvmpt3	⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑉

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fvmpt3

Step	Hyp	Ref	Expression
1		fvmpt3.a	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
2	1	eleq1d 2265	. . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑉 ↔ 𝐶 ∈ 𝑉))
3		fvmpt3.c	. . 3 ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉)
4	2, 3	vtoclga 2830	. 2 ⊢ (𝐴 ∈ 𝐷 → 𝐶 ∈ 𝑉)
5		fvmpt3.b	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
6	1, 5	fvmptg 5640	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶)
7	4, 6	mpdan 421	1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167   ↦ cmpt 4095  ‘cfv 5259

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267

This theorem is referenced by:  fvmpt3i  5644  frec2uzsucd  10510