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Theorem fvmptf 6257
 Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6237 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 𝑥𝐴
fvmptf.2 𝑥𝐶
fvmptf.3 (𝑥 = 𝐴𝐵 = 𝐶)
fvmptf.4 𝐹 = (𝑥𝐷𝐵)
Assertion
Ref Expression
fvmptf ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
Distinct variable group:   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptf
StepHypRef Expression
1 elex 3198 . . 3 (𝐶𝑉𝐶 ∈ V)
2 fvmptf.1 . . . 4 𝑥𝐴
3 fvmptf.2 . . . . . 6 𝑥𝐶
43nfel1 2775 . . . . 5 𝑥 𝐶 ∈ V
5 fvmptf.4 . . . . . . . 8 𝐹 = (𝑥𝐷𝐵)
6 nfmpt1 4707 . . . . . . . 8 𝑥(𝑥𝐷𝐵)
75, 6nfcxfr 2759 . . . . . . 7 𝑥𝐹
87, 2nffv 6155 . . . . . 6 𝑥(𝐹𝐴)
98, 3nfeq 2772 . . . . 5 𝑥(𝐹𝐴) = 𝐶
104, 9nfim 1822 . . . 4 𝑥(𝐶 ∈ V → (𝐹𝐴) = 𝐶)
11 fvmptf.3 . . . . . 6 (𝑥 = 𝐴𝐵 = 𝐶)
1211eleq1d 2683 . . . . 5 (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
13 fveq2 6148 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1413, 11eqeq12d 2636 . . . . 5 (𝑥 = 𝐴 → ((𝐹𝑥) = 𝐵 ↔ (𝐹𝐴) = 𝐶))
1512, 14imbi12d 334 . . . 4 (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹𝐴) = 𝐶)))
165fvmpt2 6248 . . . . 5 ((𝑥𝐷𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
1716ex 450 . . . 4 (𝑥𝐷 → (𝐵 ∈ V → (𝐹𝑥) = 𝐵))
182, 10, 15, 17vtoclgaf 3257 . . 3 (𝐴𝐷 → (𝐶 ∈ V → (𝐹𝐴) = 𝐶))
191, 18syl5 34 . 2 (𝐴𝐷 → (𝐶𝑉 → (𝐹𝐴) = 𝐶))
2019imp 445 1 ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Ⅎwnfc 2748  Vcvv 3186   ↦ cmpt 4673  ‘cfv 5847 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855 This theorem is referenced by:  fvmptnf  6258  elfvmptrab1  6261  elovmpt3rab1  6846  rdgsucmptf  7469  frsucmpt  7478  fprodntriv  14597  prodss  14602  fprodefsum  14750  dvfsumabs  23690  dvfsumlem1  23693  dvfsumlem4  23696  dvfsum2  23701  dchrisumlem2  25079  dchrisumlem3  25080  ptrest  33040  hlhilset  36706  fvmptd3  38922  fsumsermpt  39215  mulc1cncfg  39225  expcnfg  39227  climsubmpt  39296  climeldmeqmpt  39304  climfveqmpt  39307  fnlimfvre  39310  fnlimfvre2  39313  climfveqmpt3  39318  climeldmeqmpt3  39325  climinf2mpt  39350  climinfmpt  39351  stoweidlem23  39547  stoweidlem34  39558  stoweidlem36  39560  wallispilem5  39593  stirlinglem4  39601  stirlinglem11  39608  stirlinglem12  39609  stirlinglem13  39610  stirlinglem14  39611  sge0lempt  39934  sge0isummpt2  39956  meadjiun  39990  hoimbl2  40186  vonhoire  40193
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