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Theorem fvreseq0 6468
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 6153 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
2 fnssres 6153 . . 3 ((𝐺 Fn 𝐶𝐵𝐶) → (𝐺𝐵) Fn 𝐵)
3 eqfnfv 6462 . . . 4 (((𝐹𝐵) Fn 𝐵 ∧ (𝐺𝐵) Fn 𝐵) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥)))
4 fvres 6356 . . . . . 6 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
5 fvres 6356 . . . . . 6 (𝑥𝐵 → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
64, 5eqeq12d 2763 . . . . 5 (𝑥𝐵 → (((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥) ↔ (𝐹𝑥) = (𝐺𝑥)))
76ralbiia 3105 . . . 4 (∀𝑥𝐵 ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
83, 7syl6bb 276 . . 3 (((𝐹𝐵) Fn 𝐵 ∧ (𝐺𝐵) Fn 𝐵) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
91, 2, 8syl2an 495 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ (𝐺 Fn 𝐶𝐵𝐶)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
109an4s 904 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐵𝐴𝐵𝐶)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1620  wcel 2127  wral 3038  wss 3703  cres 5256   Fn wfn 6032  cfv 6037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-fv 6045
This theorem is referenced by:  fvreseq1  6469  fvreseq  6470  limsupequzlem  40426
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