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Theorem feqmptdf 6410
Description: Deduction form of dffn5f 6411. (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 6196 . 2 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 fnrel 6140 . . . . 5 (𝐹 Fn 𝐴 → Rel 𝐹)
4 feqmptdf.2 . . . . . 6 𝑥𝐹
5 nfcv 2916 . . . . . 6 𝑦𝐹
64, 5dfrel4 5737 . . . . 5 (Rel 𝐹𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
73, 6sylib 209 . . . 4 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
8 feqmptdf.1 . . . . . 6 𝑥𝐴
94, 8nffn 6138 . . . . 5 𝑥 𝐹 Fn 𝐴
10 nfv 1998 . . . . 5 𝑦 𝐹 Fn 𝐴
11 fnbr 6144 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
1211ex 398 . . . . . . 7 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
1312pm4.71rd 553 . . . . . 6 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑥𝐹𝑦)))
14 eqcom 2781 . . . . . . . 8 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
15 fnbrfvb 6394 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
1614, 15syl5bb 273 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
1716pm5.32da 569 . . . . . 6 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑦 = (𝐹𝑥)) ↔ (𝑥𝐴𝑥𝐹𝑦)))
1813, 17bitr4d 272 . . . . 5 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = (𝐹𝑥))))
199, 10, 18opabbid 4862 . . . 4 (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
207, 19eqtrd 2808 . . 3 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
21 df-mpt 4877 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))}
2220, 21syl6eqr 2826 . 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 383   = wceq 1634  wcel 2148  wnfc 2903   class class class wbr 4797  {copab 4859  cmpt 4876  Rel wrel 5268   Fn wfn 6037  wf 6038  cfv 6042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pr 5048
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-fv 6050
This theorem is referenced by:  esumf1o  30469  feqresmptf  39960  liminfvaluz3  40552  liminfvaluz4  40555  volioofmpt  40734  volicofmpt  40737
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