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Theorem feqmptdf 6290
Description: Deduction form of dffn5f 6291. (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 6083 . 2 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 fnrel 6027 . . . . 5 (𝐹 Fn 𝐴 → Rel 𝐹)
4 feqmptdf.2 . . . . . 6 𝑥𝐹
5 nfcv 2793 . . . . . 6 𝑦𝐹
64, 5dfrel4 5620 . . . . 5 (Rel 𝐹𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
73, 6sylib 208 . . . 4 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
8 feqmptdf.1 . . . . . 6 𝑥𝐴
94, 8nffn 6025 . . . . 5 𝑥 𝐹 Fn 𝐴
10 nfv 1883 . . . . 5 𝑦 𝐹 Fn 𝐴
11 fnbr 6031 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
1211ex 449 . . . . . . 7 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
1312pm4.71rd 668 . . . . . 6 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑥𝐹𝑦)))
14 eqcom 2658 . . . . . . . 8 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
15 fnbrfvb 6274 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
1614, 15syl5bb 272 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
1716pm5.32da 674 . . . . . 6 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑦 = (𝐹𝑥)) ↔ (𝑥𝐴𝑥𝐹𝑦)))
1813, 17bitr4d 271 . . . . 5 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = (𝐹𝑥))))
199, 10, 18opabbid 4748 . . . 4 (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
207, 19eqtrd 2685 . . 3 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
21 df-mpt 4763 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))}
2220, 21syl6eqr 2703 . 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 1523  wcel 2030  wnfc 2780   class class class wbr 4685  {copab 4745  cmpt 4762  Rel wrel 5148   Fn wfn 5921  wf 5922  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934
This theorem is referenced by:  esumf1o  30240  feqresmptf  39747  liminfvaluz3  40346  liminfvaluz4  40349  volioofmpt  40529  volicofmpt  40532
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