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Theorem fvreseq1 6482
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 6161 . . . . 5 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
21ad2antlr 765 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → (𝐺𝐵) = 𝐺)
32eqcomd 2766 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → 𝐺 = (𝐺𝐵))
43eqeq2d 2770 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = 𝐺 ↔ (𝐹𝐵) = (𝐺𝐵)))
5 ssid 3765 . . 3 𝐵𝐵
6 fvreseq0 6481 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐵𝐴𝐵𝐵)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
75, 6mpanr2 722 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
84, 7bitrd 268 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = 𝐺 ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wral 3050  wss 3715  cres 5268   Fn wfn 6044  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  symgextres  18065  sseqfres  30785
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