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Theorem dfateq12d 45511
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 5881 . . . 4 (𝜑 → dom 𝐹 = dom 𝐺)
41, 3eleq12d 2826 . . 3 (𝜑 → (𝐴 ∈ dom 𝐹𝐵 ∈ dom 𝐺))
51sneqd 4618 . . . . 5 (𝜑 → {𝐴} = {𝐵})
62, 5reseq12d 5958 . . . 4 (𝜑 → (𝐹 ↾ {𝐴}) = (𝐺 ↾ {𝐵}))
76funeqd 6543 . . 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 45504 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
10 df-dfat 45504 . 2 (𝐺 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵})))
118, 9, 103bitr4g 313 1 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {csn 4606  dom cdm 5653  cres 5655  Fun wfun 6510   defAt wdfat 45501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3419  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-br 5126  df-opab 5188  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-res 5665  df-fun 6518  df-dfat 45504
This theorem is referenced by:  afveq12d  45518  afv2eq12d  45600
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