Theorem fvmptf 6962

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6939 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 2915	. . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ V
4		fvmptf.4	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
5		nfmpt1 5197	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵)
6	4, 5	nfcxfr 2896	. . . . . 6 ⊢ Ⅎ𝑥𝐹
7	6, 1	nffv 6844	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴)
8	7, 2	nfeq 2912	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶
9	3, 8	nfim 1897	. . 3 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)
10		fvmptf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
11	10	eleq1d 2821	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
12		fveq2 6834	. . . . 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 6952	. . . 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 3531	. 2 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶))
18		elex 3461	. 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 1541   ∈ wcel 2113  Ⅎwnfc 2883  Vcvv 3440   ↦ cmpt 5179  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500

This theorem is referenced by:  fvmptnf  6963  elfvmptrab1w  6968  elfvmptrab1  6969  elovmpt3rab1  7618  rdgsucmptf  8359  frsucmpt  8369  fprodntriv  15865  prodss  15870  fprodefsum  16018  dvfsumabs  25985  dvfsumlem1  25988  dvfsumlem4  25992  dvfsum2  25997  dchrisumlem2  27457  dchrisumlem3  27458  rmfsupp2  33320  ptrest  37816  hlhilset  42190  orbitclmpt  45195  fsumsermpt  45821  mulc1cncfg  45831  expcnfg  45833  climsubmpt  45900  climeldmeqmpt  45908  climfveqmpt  45911  fnlimfvre  45914  climfveqmpt3  45922  climeldmeqmpt3  45929  climinf2mpt  45954  climinfmpt  45955  stoweidlem23  46263  stoweidlem34  46274  stoweidlem36  46276  wallispilem5  46309  stirlinglem4  46317  stirlinglem11  46324  stirlinglem12  46325  stirlinglem13  46326  stirlinglem14  46327  sge0lempt  46650  sge0isummpt2  46672  meadjiun  46706  hoimbl2  46905  vonhoire  46912