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Theorem fvreseq1 6802
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 6460 . . . . 5 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
21ad2antlr 723 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → (𝐺𝐵) = 𝐺)
32eqcomd 2827 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → 𝐺 = (𝐺𝐵))
43eqeq2d 2832 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = 𝐺 ↔ (𝐹𝐵) = (𝐺𝐵)))
5 ssid 3988 . . 3 𝐵𝐵
6 fvreseq0 6801 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐵𝐴𝐵𝐵)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
75, 6mpanr2 700 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
84, 7bitrd 280 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = 𝐺 ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wral 3138  wss 3935  cres 5551   Fn wfn 6344  cfv 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357
This theorem is referenced by:  symgextres  18484  sseqfres  31551
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