Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1502 Structured version   Unicode version

Theorem bnj1502 33202
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1502.1  |-  ( ph  ->  Fun  F )
bnj1502.2  |-  ( ph  ->  G  C_  F )
bnj1502.3  |-  ( ph  ->  A  e.  dom  G
)
Assertion
Ref Expression
bnj1502  |-  ( ph  ->  ( F `  A
)  =  ( G `
 A ) )

Proof of Theorem bnj1502
StepHypRef Expression
1 bnj1502.1 . 2  |-  ( ph  ->  Fun  F )
2 bnj1502.2 . 2  |-  ( ph  ->  G  C_  F )
3 bnj1502.3 . 2  |-  ( ph  ->  A  e.  dom  G
)
4 funssfv 5881 . 2  |-  ( ( Fun  F  /\  G  C_  F  /\  A  e. 
dom  G )  -> 
( F `  A
)  =  ( G `
 A ) )
51, 2, 3, 4syl3anc 1228 1  |-  ( ph  ->  ( F `  A
)  =  ( G `
 A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    C_ wss 3476   dom cdm 4999   Fun wfun 5582   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fv 5596
This theorem is referenced by:  bnj570  33259  bnj929  33290  bnj1450  33402  bnj1501  33419
  Copyright terms: Public domain W3C validator