Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj156 Structured version   Unicode version

Theorem bnj156 29097
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj156.1  |-  ( ze0  <->  (
f  Fn  1o  /\  ph' 
/\  ps' ) )
bnj156.2  |-  ( ze1  <->  [. g  /  f ]. ze0 )
bnj156.3  |-  ( ph1  <->  [. g  /  f ]. ph' )
bnj156.4  |-  ( ps1  <->  [. g  /  f ]. ps' )
Assertion
Ref Expression
bnj156  |-  ( ze1  <->  (
g  Fn  1o  /\  ph1 
/\  ps1 ) )

Proof of Theorem bnj156
StepHypRef Expression
1 bnj156.2 . 2  |-  ( ze1  <->  [. g  /  f ]. ze0 )
2 bnj156.1 . . . 4  |-  ( ze0  <->  (
f  Fn  1o  /\  ph' 
/\  ps' ) )
32sbcbii 3332 . . 3  |-  ( [. g  /  f ]. ze0  <->  [. g  / 
f ]. ( f  Fn  1o  /\  ph'  /\  ps' ) )
4 sbc3an 3334 . . . 4  |-  ( [. g  /  f ]. (
f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( [. g  /  f ]. f  Fn  1o  /\  [. g  /  f ]. ph'  /\  [. g  /  f ]. ps' ) )
5 bnj62 29087 . . . . 5  |-  ( [. g  /  f ]. f  Fn  1o  <->  g  Fn  1o )
6 bnj156.3 . . . . . 6  |-  ( ph1  <->  [. g  /  f ]. ph' )
76bicomi 202 . . . . 5  |-  ( [. g  /  f ]. ph'  <->  ph1 )
8 bnj156.4 . . . . . 6  |-  ( ps1  <->  [. g  /  f ]. ps' )
98bicomi 202 . . . . 5  |-  ( [. g  /  f ]. ps'  <->  ps1 )
105, 7, 93anbi123i 1186 . . . 4  |-  ( (
[. g  /  f ]. f  Fn  1o  /\ 
[. g  /  f ]. ph'  /\  [. g  /  f ]. ps' )  <->  ( g  Fn  1o  /\  ph1  /\  ps1 )
)
114, 10bitri 249 . . 3  |-  ( [. g  /  f ]. (
f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( g  Fn  1o  /\  ph1  /\  ps1 )
)
123, 11bitri 249 . 2  |-  ( [. g  /  f ]. ze0  <->  ( g  Fn  1o  /\  ph1  /\  ps1 )
)
131, 12bitri 249 1  |-  ( ze1  <->  (
g  Fn  1o  /\  ph1 
/\  ps1 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ w3a 974   [.wsbc 3276    Fn wfn 5563   1oc1o 7159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-fun 5570  df-fn 5571
This theorem is referenced by:  bnj153  29252
  Copyright terms: Public domain W3C validator