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Theorem bnj1502 29488
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 5887 . 2  |-  ( ( Fun  F  /\  G  C_  F  /\  A  e. 
dom  G )  -> 
( F `  A
)  =  ( G `
 A ) )
51, 2, 3, 4syl3anc 1264 1  |-  ( ph  ->  ( F `  A
)  =  ( G `
 A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867    C_ wss 3433   dom cdm 4845   Fun wfun 5586   ` cfv 5592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5556  df-fun 5594  df-fv 5600
This theorem is referenced by:  bnj570  29545  bnj929  29576  bnj1450  29688  bnj1501  29705
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