Theorem fvmpt3 6996

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 3537	. 2 ⊢ (𝐴 ∈ 𝐷 → 𝐶 ∈ 𝑉)
5		fvmpt3.b	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
6	1, 5	fvmptg 6989	. 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 1570   ∈ wcel 2145   ↦ cmpt 5186  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fv 6545

This theorem is used by:  fvmpt3i  6997  harval  9547  mrcfval  17775  elmptrab  24139  zringfrac  34079  frlmsnic  43584  wallispi  47049  1arymaptfv  49721  2arymaptfv  49732