Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1503 Structured version   Visualization version   GIF version

Theorem bnj1503 31436
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1503.1 (𝜑 → Fun 𝐹)
bnj1503.2 (𝜑𝐺𝐹)
bnj1503.3 (𝜑𝐴 ⊆ dom 𝐺)
Assertion
Ref Expression
bnj1503 (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Proof of Theorem bnj1503
StepHypRef Expression
1 bnj1503.1 . 2 (𝜑 → Fun 𝐹)
2 bnj1503.2 . 2 (𝜑𝐺𝐹)
3 bnj1503.3 . 2 (𝜑𝐴 ⊆ dom 𝐺)
4 fun2ssres 6145 . 2 ((Fun 𝐹𝐺𝐹𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
51, 2, 3, 4syl3anc 1491 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wss 3769  dom cdm 5312  cres 5314  Fun wfun 6095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-res 5324  df-fun 6103
This theorem is referenced by:  bnj1442  31634  bnj1450  31635  bnj1501  31652
  Copyright terms: Public domain W3C validator