ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  feqmptd GIF version

Theorem feqmptd 5357
Description: Deduction form of dffn5im 5350. (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 5161 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
31, 2syl 14 . 2 (𝜑𝐹 Fn 𝐴)
4 dffn5im 5350 . 2 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
53, 4syl 14 1 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  cmpt 3899   Fn wfn 5010  wf 5011  cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023
This theorem is referenced by:  feqresmpt  5358  fcoconst  5468  suppssof1  5872  ofco  5873  caofinvl  5877  caofcom  5878  mapxpen  6564  xpmapenlem  6565  cnrecnv  10344
  Copyright terms: Public domain W3C validator