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Theorem fvreseq0 7058
Description: Equality of restricted functions is determined by their values (for functions with different domains). (Contributed by AV, 6-Jan-2019.)
Assertion
Ref Expression
fvreseq0 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐵𝐴𝐵𝐶)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem fvreseq0
StepHypRef Expression
1 fnssres 6691 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
2 fnssres 6691 . . 3 ((𝐺 Fn 𝐶𝐵𝐶) → (𝐺𝐵) Fn 𝐵)
3 eqfnfv 7051 . . . 4 (((𝐹𝐵) Fn 𝐵 ∧ (𝐺𝐵) Fn 𝐵) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥)))
4 fvres 6925 . . . . . 6 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
5 fvres 6925 . . . . . 6 (𝑥𝐵 → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
64, 5eqeq12d 2753 . . . . 5 (𝑥𝐵 → (((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥) ↔ (𝐹𝑥) = (𝐺𝑥)))
76ralbiia 3091 . . . 4 (∀𝑥𝐵 ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
83, 7bitrdi 287 . . 3 (((𝐹𝐵) Fn 𝐵 ∧ (𝐺𝐵) Fn 𝐵) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
91, 2, 8syl2an 596 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ (𝐺 Fn 𝐶𝐵𝐶)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
109an4s 660 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐵𝐴𝐵𝐶)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wss 3951  cres 5687   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  fvreseq1  7059  fvreseq  7060  ply1degltdimlem  33673  limsupequzlem  45737
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