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Theorem fvmptf 6340
Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6319 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 3243 . . 3 (𝐶𝑉𝐶 ∈ V)
2 fvmptf.1 . . . 4 𝑥𝐴
3 fvmptf.2 . . . . . 6 𝑥𝐶
43nfel1 2808 . . . . 5 𝑥 𝐶 ∈ V
5 fvmptf.4 . . . . . . . 8 𝐹 = (𝑥𝐷𝐵)
6 nfmpt1 4780 . . . . . . . 8 𝑥(𝑥𝐷𝐵)
75, 6nfcxfr 2791 . . . . . . 7 𝑥𝐹
87, 2nffv 6236 . . . . . 6 𝑥(𝐹𝐴)
98, 3nfeq 2805 . . . . 5 𝑥(𝐹𝐴) = 𝐶
104, 9nfim 1865 . . . 4 𝑥(𝐶 ∈ V → (𝐹𝐴) = 𝐶)
11 fvmptf.3 . . . . . 6 (𝑥 = 𝐴𝐵 = 𝐶)
1211eleq1d 2715 . . . . 5 (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
13 fveq2 6229 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1413, 11eqeq12d 2666 . . . . 5 (𝑥 = 𝐴 → ((𝐹𝑥) = 𝐵 ↔ (𝐹𝐴) = 𝐶))
1512, 14imbi12d 333 . . . 4 (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹𝐴) = 𝐶)))
165fvmpt2 6330 . . . . 5 ((𝑥𝐷𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
1716ex 449 . . . 4 (𝑥𝐷 → (𝐵 ∈ V → (𝐹𝑥) = 𝐵))
182, 10, 15, 17vtoclgaf 3302 . . 3 (𝐴𝐷 → (𝐶 ∈ V → (𝐹𝐴) = 𝐶))
191, 18syl5 34 . 2 (𝐴𝐷 → (𝐶𝑉 → (𝐹𝐴) = 𝐶))
2019imp 444 1 ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wnfc 2780  Vcvv 3231  cmpt 4762  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  fvmptnf  6341  elfvmptrab1  6344  elovmpt3rab1  6935  rdgsucmptf  7569  frsucmpt  7578  fprodntriv  14716  prodss  14721  fprodefsum  14869  dvfsumabs  23831  dvfsumlem1  23834  dvfsumlem4  23837  dvfsum2  23842  dchrisumlem2  25224  dchrisumlem3  25225  ptrest  33538  hlhilset  37543  fvmptd3  39761  fsumsermpt  40129  mulc1cncfg  40139  expcnfg  40141  climsubmpt  40210  climeldmeqmpt  40218  climfveqmpt  40221  fnlimfvre  40224  fnlimfvre2  40227  climfveqmpt3  40232  climeldmeqmpt3  40239  climinf2mpt  40264  climinfmpt  40265  stoweidlem23  40558  stoweidlem34  40569  stoweidlem36  40571  wallispilem5  40604  stirlinglem4  40612  stirlinglem11  40619  stirlinglem12  40620  stirlinglem13  40621  stirlinglem14  40622  sge0lempt  40945  sge0isummpt2  40967  meadjiun  41001  hoimbl2  41200  vonhoire  41207
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