Theorem fvmptf 7012

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6989 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
fvmptf.1	⊢ Ⅎ𝑥𝐴
fvmptf.2	⊢ Ⅎ𝑥𝐶
fvmptf.3	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
fvmptf.4	⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)

Assertion

Ref	Expression
fvmptf	⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶)

Distinct variable group:   𝑥,𝐷

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptf

Step	Hyp	Ref	Expression
1		fvmptf.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		fvmptf.2	. . . . 5 ⊢ Ⅎ𝑥𝐶
3	2	nfel1 2916	. . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ V
4		fvmptf.4	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
5		nfmpt1 5225	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵)
6	4, 5	nfcxfr 2897	. . . . . 6 ⊢ Ⅎ𝑥𝐹
7	6, 1	nffv 6891	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴)
8	7, 2	nfeq 2913	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶
9	3, 8	nfim 1896	. . 3 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)
10		fvmptf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
11	10	eleq1d 2820	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
12		fveq2 6881	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
13	12, 10	eqeq12d 2752	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝐴) = 𝐶))
14	11, 13	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹‘𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)))
15	4	fvmpt2 7002	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵)
16	15	ex 412	. . 3 ⊢ (𝑥 ∈ 𝐷 → (𝐵 ∈ V → (𝐹‘𝑥) = 𝐵))
17	1, 9, 14, 16	vtoclgaf 3560	. 2 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶))
18		elex 3485	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
19	17, 18	impel 505	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Ⅎwnfc 2884  Vcvv 3464   ↦ cmpt 5206  ‘cfv 6536

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fv 6544

This theorem is referenced by:  fvmptnf  7013  elfvmptrab1w  7018  elfvmptrab1  7019  elovmpt3rab1  7672  rdgsucmptf  8447  frsucmpt  8457  fprodntriv  15963  prodss  15968  fprodefsum  16116  dvfsumabs  25986  dvfsumlem1  25989  dvfsumlem4  25993  dvfsum2  25998  dchrisumlem2  27458  dchrisumlem3  27459  rmfsupp2  33238  ptrest  37648  hlhilset  41958  orbitclmpt  44950  fsumsermpt  45575  mulc1cncfg  45585  expcnfg  45587  climsubmpt  45656  climeldmeqmpt  45664  climfveqmpt  45667  fnlimfvre  45670  climfveqmpt3  45678  climeldmeqmpt3  45685  climinf2mpt  45710  climinfmpt  45711  stoweidlem23  46019  stoweidlem34  46030  stoweidlem36  46032  wallispilem5  46065  stirlinglem4  46073  stirlinglem11  46080  stirlinglem12  46081  stirlinglem13  46082  stirlinglem14  46083  sge0lempt  46406  sge0isummpt2  46428  meadjiun  46462  hoimbl2  46661  vonhoire  46668