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Theorem dfateq12d 39752
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 5139 . . . 4 (𝜑 → dom 𝐹 = dom 𝐺)
41, 3eleq12d 2586 . . 3 (𝜑 → (𝐴 ∈ dom 𝐹𝐵 ∈ dom 𝐺))
51sneqd 4040 . . . . 5 (𝜑 → {𝐴} = {𝐵})
62, 5reseq12d 5209 . . . 4 (𝜑 → (𝐹 ↾ {𝐴}) = (𝐺 ↾ {𝐵}))
76funeqd 5710 . . 3 (𝜑 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun (𝐺 ↾ {𝐵})))
84, 7anbi12d 742 . 2 (𝜑 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵}))))
9 df-dfat 39738 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
10 df-dfat 39738 . 2 (𝐺 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵})))
118, 9, 103bitr4g 301 1 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1938  {csn 4028  dom cdm 4932  cres 4934  Fun wfun 5683   defAt wdfat 39735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-br 4482  df-opab 4542  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-res 4944  df-fun 5691  df-dfat 39738
This theorem is referenced by:  afveq12d  39756
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