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Theorem dfateq12d 40543
Description: Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
dfateq12d.1 (𝜑𝐹 = 𝐺)
dfateq12d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
dfateq12d (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))

Proof of Theorem dfateq12d
StepHypRef Expression
1 dfateq12d.2 . . . 4 (𝜑𝐴 = 𝐵)
2 dfateq12d.1 . . . . 5 (𝜑𝐹 = 𝐺)
32dmeqd 5296 . . . 4 (𝜑 → dom 𝐹 = dom 𝐺)
41, 3eleq12d 2692 . . 3 (𝜑 → (𝐴 ∈ dom 𝐹𝐵 ∈ dom 𝐺))
51sneqd 4167 . . . . 5 (𝜑 → {𝐴} = {𝐵})
62, 5reseq12d 5367 . . . 4 (𝜑 → (𝐹 ↾ {𝐴}) = (𝐺 ↾ {𝐵}))
76funeqd 5879 . . 3 (𝜑 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun (𝐺 ↾ {𝐵})))
84, 7anbi12d 746 . 2 (𝜑 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵}))))
9 df-dfat 40530 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
10 df-dfat 40530 . 2 (𝐺 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵})))
118, 9, 103bitr4g 303 1 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {csn 4155  dom cdm 5084  cres 5086  Fun wfun 5851   defAt wdfat 40527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-res 5096  df-fun 5859  df-dfat 40530
This theorem is referenced by:  afveq12d  40547
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