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Theorem fvreseq1 6859
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 6496 . . . . 5 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
21ad2antlr 727 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → (𝐺𝐵) = 𝐺)
32eqcomd 2743 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → 𝐺 = (𝐺𝐵))
43eqeq2d 2748 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = 𝐺 ↔ (𝐹𝐵) = (𝐺𝐵)))
5 ssid 3923 . . 3 𝐵𝐵
6 fvreseq0 6858 . . 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 282 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = 𝐺 ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wral 3061  wss 3866  cres 5553   Fn wfn 6375  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388
This theorem is referenced by:  symgextres  18817  sseqfres  32072
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