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Theorem feqmptdf 6908
Description: Deduction form of dffn5f 6909. (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 6664 . 2 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 fnrel 6600 . . . . 5 (𝐹 Fn 𝐴 → Rel 𝐹)
4 feqmptdf.2 . . . . . 6 𝑥𝐹
5 nfcv 2906 . . . . . 6 𝑦𝐹
64, 5dfrel4 6140 . . . . 5 (Rel 𝐹𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
73, 6sylib 217 . . . 4 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
8 feqmptdf.1 . . . . . 6 𝑥𝐴
94, 8nffn 6597 . . . . 5 𝑥 𝐹 Fn 𝐴
10 nfv 1918 . . . . 5 𝑦 𝐹 Fn 𝐴
11 fnbr 6606 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
1211ex 414 . . . . . . 7 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
1312pm4.71rd 564 . . . . . 6 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑥𝐹𝑦)))
14 eqcom 2745 . . . . . . . 8 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
15 fnbrfvb 6891 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
1614, 15bitrid 283 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
1716pm5.32da 580 . . . . . 6 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑦 = (𝐹𝑥)) ↔ (𝑥𝐴𝑥𝐹𝑦)))
1813, 17bitr4d 282 . . . . 5 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = (𝐹𝑥))))
199, 10, 18opabbid 5169 . . . 4 (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
207, 19eqtrd 2778 . . 3 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
21 df-mpt 5188 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))}
2220, 21eqtr4di 2796 . 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 397   = wceq 1542  wcel 2107  wnfc 2886   class class class wbr 5104  {copab 5166  cmpt 5187  Rel wrel 5636   Fn wfn 6487  wf 6488  cfv 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500
This theorem is referenced by:  esumf1o  32410  feqresmptf  43180  liminfvaluz3  43759  liminfvaluz4  43762  volioofmpt  43957  volicofmpt  43960
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