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Theorem dfateq12d 42149
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 5571 . . . 4 (𝜑 → dom 𝐹 = dom 𝐺)
41, 3eleq12d 2852 . . 3 (𝜑 → (𝐴 ∈ dom 𝐹𝐵 ∈ dom 𝐺))
51sneqd 4409 . . . . 5 (𝜑 → {𝐴} = {𝐵})
62, 5reseq12d 5643 . . . 4 (𝜑 → (𝐹 ↾ {𝐴}) = (𝐺 ↾ {𝐵}))
76funeqd 6157 . . 3 (𝜑 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun (𝐺 ↾ {𝐵})))
84, 7anbi12d 624 . 2 (𝜑 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵}))))
9 df-dfat 42142 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
10 df-dfat 42142 . 2 (𝐺 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵})))
118, 9, 103bitr4g 306 1 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2106  {csn 4397  dom cdm 5355  cres 5357  Fun wfun 6129   defAt wdfat 42139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-res 5367  df-fun 6137  df-dfat 42142
This theorem is referenced by:  afveq12d  42156  afv2eq12d  42238
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