Theorem fvmptf 7022

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6999 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 2909	. . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ V
4		fvmptf.4	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
5		nfmpt1 5253	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵)
6	4, 5	nfcxfr 2890	. . . . . 6 ⊢ Ⅎ𝑥𝐹
7	6, 1	nffv 6903	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴)
8	7, 2	nfeq 2906	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶
9	3, 8	nfim 1892	. . 3 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)
10		fvmptf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
11	10	eleq1d 2811	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
12		fveq2 6893	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
13	12, 10	eqeq12d 2742	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝐴) = 𝐶))
14	11, 13	imbi12d 343	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹‘𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)))
15	4	fvmpt2 7012	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵)
16	15	ex 411	. . 3 ⊢ (𝑥 ∈ 𝐷 → (𝐵 ∈ V → (𝐹‘𝑥) = 𝐵))
17	1, 9, 14, 16	vtoclgaf 3556	. 2 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶))
18		elex 3482	. 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
19	17, 18	impel 504	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  Ⅎwnfc 2876  Vcvv 3462   ↦ cmpt 5228  ‘cfv 6546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fv 6554

This theorem is referenced by:  fvmptnf  7023  elfvmptrab1w  7028  elfvmptrab1  7029  elovmpt3rab1  7678  rdgsucmptf  8450  frsucmpt  8460  fprodntriv  15939  prodss  15944  fprodefsum  16092  dvfsumabs  26045  dvfsumlem1  26048  dvfsumlem4  26052  dvfsum2  26057  dchrisumlem2  27516  dchrisumlem3  27517  rmfsupp2  33108  ptrest  37333  hlhilset  41646  fsumsermpt  45236  mulc1cncfg  45246  expcnfg  45248  climsubmpt  45317  climeldmeqmpt  45325  climfveqmpt  45328  fnlimfvre  45331  climfveqmpt3  45339  climeldmeqmpt3  45346  climinf2mpt  45371  climinfmpt  45372  stoweidlem23  45680  stoweidlem34  45691  stoweidlem36  45693  wallispilem5  45726  stirlinglem4  45734  stirlinglem11  45741  stirlinglem12  45742  stirlinglem13  45743  stirlinglem14  45744  sge0lempt  46067  sge0isummpt2  46089  meadjiun  46123  hoimbl2  46322  vonhoire  46329