MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvreseq0 Structured version   Visualization version   GIF version

Theorem fvreseq0 7033
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 6667 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
2 fnssres 6667 . . 3 ((𝐺 Fn 𝐶𝐵𝐶) → (𝐺𝐵) Fn 𝐵)
3 eqfnfv 7026 . . . 4 (((𝐹𝐵) Fn 𝐵 ∧ (𝐺𝐵) Fn 𝐵) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥)))
4 fvres 6904 . . . . . 6 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
5 fvres 6904 . . . . . 6 (𝑥𝐵 → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
64, 5eqeq12d 2742 . . . . 5 (𝑥𝐵 → (((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥) ↔ (𝐹𝑥) = (𝐺𝑥)))
76ralbiia 3085 . . . 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 595 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ (𝐺 Fn 𝐶𝐵𝐶)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
109an4s 657 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐵𝐴𝐵𝐶)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  wss 3943  cres 5671   Fn wfn 6532  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-fv 6545
This theorem is referenced by:  fvreseq1  7034  fvreseq  7035  ply1degltdimlem  33225  limsupequzlem  45010
  Copyright terms: Public domain W3C validator