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Theorem fvreseq1 7059
Description: Equality of a function restricted to the domain of another function. (Contributed by AV, 6-Jan-2019.)
Assertion
Ref Expression
fvreseq1 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = 𝐺 ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem fvreseq1
StepHypRef Expression
1 fnresdm 6688 . . . . 5 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
21ad2antlr 727 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → (𝐺𝐵) = 𝐺)
32eqcomd 2741 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → 𝐺 = (𝐺𝐵))
43eqeq2d 2746 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = 𝐺 ↔ (𝐹𝐵) = (𝐺𝐵)))
5 ssid 4018 . . 3 𝐵𝐵
6 fvreseq0 7058 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐵𝐴𝐵𝐵)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
75, 6mpanr2 704 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
84, 7bitrd 279 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = 𝐺 ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wral 3059  wss 3963  cres 5691   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  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-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  symgextres  19458  ressply1evl  22390  sseqfres  34375
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