MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvreseq1 Structured version   Visualization version   GIF version

Theorem fvreseq1 6358
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 6038 . . . . 5 (𝐺 Fn 𝐵 → (𝐺𝐵) = 𝐺)
21ad2antlr 763 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → (𝐺𝐵) = 𝐺)
32eqcomd 2657 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → 𝐺 = (𝐺𝐵))
43eqeq2d 2661 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = 𝐺 ↔ (𝐹𝐵) = (𝐺𝐵)))
5 ssid 3657 . . 3 𝐵𝐵
6 fvreseq0 6357 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐵𝐴𝐵𝐵)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
75, 6mpanr2 720 . 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 1523  wral 2941  wss 3607  cres 5145   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  symgextres  17891  sseqfres  30583
  Copyright terms: Public domain W3C validator