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Theorem fvmptf 6463
Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6442 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 3352 . . 3 (𝐶𝑉𝐶 ∈ V)
2 fvmptf.1 . . . 4 𝑥𝐴
3 fvmptf.2 . . . . . 6 𝑥𝐶
43nfel1 2917 . . . . 5 𝑥 𝐶 ∈ V
5 fvmptf.4 . . . . . . . 8 𝐹 = (𝑥𝐷𝐵)
6 nfmpt1 4899 . . . . . . . 8 𝑥(𝑥𝐷𝐵)
75, 6nfcxfr 2900 . . . . . . 7 𝑥𝐹
87, 2nffv 6359 . . . . . 6 𝑥(𝐹𝐴)
98, 3nfeq 2914 . . . . 5 𝑥(𝐹𝐴) = 𝐶
104, 9nfim 1974 . . . 4 𝑥(𝐶 ∈ V → (𝐹𝐴) = 𝐶)
11 fvmptf.3 . . . . . 6 (𝑥 = 𝐴𝐵 = 𝐶)
1211eleq1d 2824 . . . . 5 (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
13 fveq2 6352 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1413, 11eqeq12d 2775 . . . . 5 (𝑥 = 𝐴 → ((𝐹𝑥) = 𝐵 ↔ (𝐹𝐴) = 𝐶))
1512, 14imbi12d 333 . . . 4 (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹𝐴) = 𝐶)))
165fvmpt2 6453 . . . . 5 ((𝑥𝐷𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
1716ex 449 . . . 4 (𝑥𝐷 → (𝐵 ∈ V → (𝐹𝑥) = 𝐵))
182, 10, 15, 17vtoclgaf 3411 . . 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 1632  wcel 2139  wnfc 2889  Vcvv 3340  cmpt 4881  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057
This theorem is referenced by:  fvmptnf  6464  elfvmptrab1  6467  elovmpt3rab1  7058  rdgsucmptf  7693  frsucmpt  7702  fprodntriv  14871  prodss  14876  fprodefsum  15024  dvfsumabs  23985  dvfsumlem1  23988  dvfsumlem4  23991  dvfsum2  23996  dchrisumlem2  25378  dchrisumlem3  25379  ptrest  33721  hlhilset  37728  fvmptd3  39946  fsumsermpt  40314  mulc1cncfg  40324  expcnfg  40326  climsubmpt  40395  climeldmeqmpt  40403  climfveqmpt  40406  fnlimfvre  40409  fnlimfvre2  40412  climfveqmpt3  40417  climeldmeqmpt3  40424  climinf2mpt  40449  climinfmpt  40450  stoweidlem23  40743  stoweidlem34  40754  stoweidlem36  40756  wallispilem5  40789  stirlinglem4  40797  stirlinglem11  40804  stirlinglem12  40805  stirlinglem13  40806  stirlinglem14  40807  sge0lempt  41130  sge0isummpt2  41152  meadjiun  41186  hoimbl2  41385  vonhoire  41392
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