Theorem fvmpt3 6974

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 2846	. . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑉 ↔ 𝐶 ∈ 𝑉))
3		fvmpt3.c	. . 3 ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉)
4	2, 3	vtoclga 3540	. 2 ⊢ (𝐴 ∈ 𝐷 → 𝐶 ∈ 𝑉)
5		fvmpt3.b	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
6	1, 5	fvmptg 6967	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶)
7	4, 6	mpdan 697	1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141   ↦ cmpt 5178  ‘cfv 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6471  df-fun 6517  df-fv 6523

This theorem is referenced by:  fvmpt3i  6975  harval  9501  mrcfval  17630  elmptrab  23874  zringfrac  33710  frlmsnic  43118  wallispi  46604  1arymaptfv  49222  2arymaptfv  49233