Theorem fvmpt3 6975

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 2814	. . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑉 ↔ 𝐶 ∈ 𝑉))
3		fvmpt3.c	. . 3 ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉)
4	2, 3	vtoclga 3546	. 2 ⊢ (𝐴 ∈ 𝐷 → 𝐶 ∈ 𝑉)
5		fvmpt3.b	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
6	1, 5	fvmptg 6969	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶)
7	4, 6	mpdan 687	1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ↦ cmpt 5191  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522

This theorem is referenced by:  fvmpt3i  6976  harval  9520  mrcfval  17576  elmptrab  23721  zringfrac  33532  frlmsnic  42535  wallispi  46075  1arymaptfv  48633  2arymaptfv  48644