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Theorem feqmptdf 6714
Description: Deduction form of dffn5f 6715. (Contributed by Mario Carneiro, 8-Jan-2015.) (Revised by Thierry Arnoux, 10-May-2017.)
Hypotheses
Ref Expression
feqmptdf.1 𝑥𝐴
feqmptdf.2 𝑥𝐹
feqmptdf.3 (𝜑𝐹:𝐴𝐵)
Assertion
Ref Expression
feqmptdf (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))

Proof of Theorem feqmptdf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 feqmptdf.3 . 2 (𝜑𝐹:𝐴𝐵)
2 ffn 6491 . 2 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 fnrel 6428 . . . . 5 (𝐹 Fn 𝐴 → Rel 𝐹)
4 feqmptdf.2 . . . . . 6 𝑥𝐹
5 nfcv 2958 . . . . . 6 𝑦𝐹
64, 5dfrel4 6019 . . . . 5 (Rel 𝐹𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
73, 6sylib 221 . . . 4 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
8 feqmptdf.1 . . . . . 6 𝑥𝐴
94, 8nffn 6426 . . . . 5 𝑥 𝐹 Fn 𝐴
10 nfv 1915 . . . . 5 𝑦 𝐹 Fn 𝐴
11 fnbr 6434 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
1211ex 416 . . . . . . 7 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
1312pm4.71rd 566 . . . . . 6 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑥𝐹𝑦)))
14 eqcom 2808 . . . . . . . 8 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
15 fnbrfvb 6697 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
1614, 15syl5bb 286 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
1716pm5.32da 582 . . . . . 6 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑦 = (𝐹𝑥)) ↔ (𝑥𝐴𝑥𝐹𝑦)))
1813, 17bitr4d 285 . . . . 5 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = (𝐹𝑥))))
199, 10, 18opabbid 5098 . . . 4 (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
207, 19eqtrd 2836 . . 3 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
21 df-mpt 5114 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))}
2220, 21eqtr4di 2854 . 2 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
231, 2, 223syl 18 1 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wnfc 2939   class class class wbr 5033  {copab 5095  cmpt 5113  Rel wrel 5528   Fn wfn 6323  wf 6324  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336
This theorem is referenced by:  esumf1o  31423  feqresmptf  41858  liminfvaluz3  42431  liminfvaluz4  42434  volioofmpt  42629  volicofmpt  42632
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