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Theorem fvmptf 6787
Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6765 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 fvmptf.1 . . 3 𝑥𝐴
2 fvmptf.2 . . . . 5 𝑥𝐶
32nfel1 2999 . . . 4 𝑥 𝐶 ∈ V
4 fvmptf.4 . . . . . . 7 𝐹 = (𝑥𝐷𝐵)
5 nfmpt1 5161 . . . . . . 7 𝑥(𝑥𝐷𝐵)
64, 5nfcxfr 2980 . . . . . 6 𝑥𝐹
76, 1nffv 6679 . . . . 5 𝑥(𝐹𝐴)
87, 2nfeq 2996 . . . 4 𝑥(𝐹𝐴) = 𝐶
93, 8nfim 1890 . . 3 𝑥(𝐶 ∈ V → (𝐹𝐴) = 𝐶)
10 fvmptf.3 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
1110eleq1d 2902 . . . 4 (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
12 fveq2 6669 . . . . 5 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1312, 10eqeq12d 2842 . . . 4 (𝑥 = 𝐴 → ((𝐹𝑥) = 𝐵 ↔ (𝐹𝐴) = 𝐶))
1411, 13imbi12d 346 . . 3 (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹𝐴) = 𝐶)))
154fvmpt2 6777 . . . 4 ((𝑥𝐷𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
1615ex 413 . . 3 (𝑥𝐷 → (𝐵 ∈ V → (𝐹𝑥) = 𝐵))
171, 9, 14, 16vtoclgaf 3578 . 2 (𝐴𝐷 → (𝐶 ∈ V → (𝐹𝐴) = 𝐶))
18 elex 3518 . 2 (𝐶𝑉𝐶 ∈ V)
1917, 18impel 506 1 ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wnfc 2966  Vcvv 3500  cmpt 5143  cfv 6354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fv 6362
This theorem is referenced by:  fvmptnf  6788  elfvmptrab1w  6792  elfvmptrab1  6793  elovmpt3rab1  7399  rdgsucmptf  8060  frsucmpt  8069  fprodntriv  15291  prodss  15296  fprodefsum  15443  dvfsumabs  24554  dvfsumlem1  24557  dvfsumlem4  24560  dvfsum2  24565  dchrisumlem2  25999  dchrisumlem3  26000  rmfsupp2  30799  ptrest  34777  hlhilset  38956  fsumsermpt  41744  mulc1cncfg  41754  expcnfg  41756  climsubmpt  41825  climeldmeqmpt  41833  climfveqmpt  41836  fnlimfvre  41839  climfveqmpt3  41847  climeldmeqmpt3  41854  climinf2mpt  41879  climinfmpt  41880  stoweidlem23  42193  stoweidlem34  42204  stoweidlem36  42206  wallispilem5  42239  stirlinglem4  42247  stirlinglem11  42254  stirlinglem12  42255  stirlinglem13  42256  stirlinglem14  42257  sge0lempt  42577  sge0isummpt2  42599  meadjiun  42633  hoimbl2  42832  vonhoire  42839
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