Theorem fvmpt3 6999

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 2845	. . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑉 ↔ 𝐶 ∈ 𝑉))
3		fvmpt3.c	. . 3 ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉)
4	2, 3	vtoclga 3536	. 2 ⊢ (𝐴 ∈ 𝐷 → 𝐶 ∈ 𝑉)
5		fvmpt3.b	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
6	1, 5	fvmptg 6992	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶)
7	4, 6	mpdan 700	1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ↦ cmpt 5185  ‘cfv 6539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547

This theorem is used by:  fvmpt3i  7000  harval  9558  mrcfval  17788  elmptrab  24152  zringfrac  34133  frlmsnic  43698  wallispi  47174  1arymaptfv  49846  2arymaptfv  49857