MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  feqmptdf Structured version   Visualization version   GIF version

Theorem feqmptdf 6907
Description: Deduction form of dffn5f 6908. (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 6663 . 2 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 fnrel 6599 . . . . 5 (𝐹 Fn 𝐴 → Rel 𝐹)
4 feqmptdf.2 . . . . . 6 𝑥𝐹
5 nfcv 2905 . . . . . 6 𝑦𝐹
64, 5dfrel4 6139 . . . . 5 (Rel 𝐹𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
73, 6sylib 217 . . . 4 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
8 feqmptdf.1 . . . . . 6 𝑥𝐴
94, 8nffn 6596 . . . . 5 𝑥 𝐹 Fn 𝐴
10 nfv 1917 . . . . 5 𝑦 𝐹 Fn 𝐴
11 fnbr 6605 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
1211ex 413 . . . . . . 7 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
1312pm4.71rd 563 . . . . . 6 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑥𝐹𝑦)))
14 eqcom 2744 . . . . . . . 8 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
15 fnbrfvb 6890 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
1614, 15bitrid 282 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
1716pm5.32da 579 . . . . . 6 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑦 = (𝐹𝑥)) ↔ (𝑥𝐴𝑥𝐹𝑦)))
1813, 17bitr4d 281 . . . . 5 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = (𝐹𝑥))))
199, 10, 18opabbid 5168 . . . 4 (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
207, 19eqtrd 2777 . . 3 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
21 df-mpt 5187 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))}
2220, 21eqtr4di 2795 . 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 396   = wceq 1541  wcel 2106  wnfc 2885   class class class wbr 5103  {copab 5165  cmpt 5186  Rel wrel 5635   Fn wfn 6486  wf 6487  cfv 6491
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5528  df-xp 5636  df-rel 5637  df-cnv 5638  df-co 5639  df-dm 5640  df-iota 6443  df-fun 6493  df-fn 6494  df-f 6495  df-fv 6499
This theorem is referenced by:  esumf1o  32422  feqresmptf  43211  liminfvaluz3  43790  liminfvaluz4  43793  volioofmpt  43988  volicofmpt  43991
  Copyright terms: Public domain W3C validator