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Theorem dfateq12d 47033
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 5914 . . . 4 (𝜑 → dom 𝐹 = dom 𝐺)
41, 3eleq12d 2831 . . 3 (𝜑 → (𝐴 ∈ dom 𝐹𝐵 ∈ dom 𝐺))
51sneqd 4643 . . . . 5 (𝜑 → {𝐴} = {𝐵})
62, 5reseq12d 5996 . . . 4 (𝜑 → (𝐹 ↾ {𝐴}) = (𝐺 ↾ {𝐵}))
76funeqd 6586 . . 3 (𝜑 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun (𝐺 ↾ {𝐵})))
84, 7anbi12d 631 . 2 (𝜑 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵}))))
9 df-dfat 47026 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
10 df-dfat 47026 . 2 (𝐺 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵})))
118, 9, 103bitr4g 314 1 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1535  wcel 2104  {csn 4631  dom cdm 5684  cres 5686  Fun wfun 6553   defAt wdfat 47023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5151  df-opab 5213  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-fun 6561  df-dfat 47026
This theorem is referenced by:  afveq12d  47040  afv2eq12d  47122
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