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Theorem fvmptf 5628
Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5612 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 2763 . . 3 (𝐶𝑉𝐶 ∈ V)
2 fvmptf.1 . . . 4 𝑥𝐴
3 fvmptf.2 . . . . . 6 𝑥𝐶
43nfel1 2343 . . . . 5 𝑥 𝐶 ∈ V
5 fvmptf.4 . . . . . . . 8 𝐹 = (𝑥𝐷𝐵)
6 nfmpt1 4111 . . . . . . . 8 𝑥(𝑥𝐷𝐵)
75, 6nfcxfr 2329 . . . . . . 7 𝑥𝐹
87, 2nffv 5544 . . . . . 6 𝑥(𝐹𝐴)
98, 3nfeq 2340 . . . . 5 𝑥(𝐹𝐴) = 𝐶
104, 9nfim 1583 . . . 4 𝑥(𝐶 ∈ V → (𝐹𝐴) = 𝐶)
11 fvmptf.3 . . . . . 6 (𝑥 = 𝐴𝐵 = 𝐶)
1211eleq1d 2258 . . . . 5 (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
13 fveq2 5534 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1413, 11eqeq12d 2204 . . . . 5 (𝑥 = 𝐴 → ((𝐹𝑥) = 𝐵 ↔ (𝐹𝐴) = 𝐶))
1512, 14imbi12d 234 . . . 4 (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹𝐴) = 𝐶)))
165fvmpt2 5619 . . . . 5 ((𝑥𝐷𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
1716ex 115 . . . 4 (𝑥𝐷 → (𝐵 ∈ V → (𝐹𝑥) = 𝐵))
182, 10, 15, 17vtoclgaf 2817 . . 3 (𝐴𝐷 → (𝐶 ∈ V → (𝐹𝐴) = 𝐶))
191, 18syl5 32 . 2 (𝐴𝐷 → (𝐶𝑉 → (𝐹𝐴) = 𝐶))
2019imp 124 1 ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wnfc 2319  Vcvv 2752  cmpt 4079  cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243
This theorem is referenced by:  fvmptd3  5629  elfvmptrab1  5630  sumrbdclem  11416  fsum3  11426  isumss  11430  prodrbdclem  11610  prodmodclem2a  11615  zproddc  11618  fprodntrivap  11623  prodssdc  11628  pcmpt  12374
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