Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1536 Structured version   Unicode version

Theorem bnj1536 33645
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1536.1  |-  ( ph  ->  F  Fn  A )
bnj1536.2  |-  ( ph  ->  G  Fn  A )
bnj1536.3  |-  ( ph  ->  B  C_  A )
bnj1536.4  |-  ( ph  ->  A. x  e.  B  ( F `  x )  =  ( G `  x ) )
Assertion
Ref Expression
bnj1536  |-  ( ph  ->  ( F  |`  B )  =  ( G  |`  B ) )
Distinct variable groups:    x, B    x, F    x, G
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem bnj1536
StepHypRef Expression
1 bnj1536.4 . 2  |-  ( ph  ->  A. x  e.  B  ( F `  x )  =  ( G `  x ) )
2 bnj1536.1 . . 3  |-  ( ph  ->  F  Fn  A )
3 bnj1536.2 . . 3  |-  ( ph  ->  G  Fn  A )
4 bnj1536.3 . . 3  |-  ( ph  ->  B  C_  A )
5 fvreseq 5974 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
62, 3, 4, 5syl21anc 1228 . 2  |-  ( ph  ->  ( ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
71, 6mpbird 232 1  |-  ( ph  ->  ( F  |`  B )  =  ( G  |`  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1383   A.wral 2793    C_ wss 3461    |` cres 4991    Fn wfn 5573   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-fv 5586
This theorem is referenced by:  bnj1523  33860
  Copyright terms: Public domain W3C validator