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Theorem feqmptd 5440
Description: Deduction form of dffn5im 5433. (Contributed by Mario Carneiro, 8-Jan-2015.)
Hypothesis
Ref Expression
feqmptd.1 (𝜑𝐹:𝐴𝐵)
Assertion
Ref Expression
feqmptd (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem feqmptd
StepHypRef Expression
1 feqmptd.1 . . 3 (𝜑𝐹:𝐴𝐵)
2 ffn 5240 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
31, 2syl 14 . 2 (𝜑𝐹 Fn 𝐴)
4 dffn5im 5433 . 2 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
53, 4syl 14 1 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  cmpt 3957   Fn wfn 5086  wf 5087  cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099
This theorem is referenced by:  feqresmpt  5441  cofmpt  5555  fcoconst  5557  suppssof1  5965  ofco  5966  caofinvl  5970  caofcom  5971  mapxpen  6708  xpmapenlem  6709  cnrecnv  10633  lmcn2  12355  cnmpt11f  12359  cnmpt21f  12367  cncfmpt1f  12659  negfcncf  12664  cnrehmeocntop  12668  dvcnp2cntop  12738  dvimulf  12745  dvcoapbr  12746  dvcj  12748  dvfre  12749  dvef  12762
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