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Theorem tfrlem2 6573
Description: Lemma for transfinite recursion. This provides some messy details needed to link tfrlem1 6572 into the main proof. (Contributed by NM, 23-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Assertion
Ref Expression
tfrlem2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G )  -> 
( A  e.  On  ->  ( A. w ( A  e.  On  ->  ( w  e.  A  -> 
( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) )  ->  y  =  z ) ) ) )
Distinct variable groups:    w, A    w, F    w, G    x, w
Allowed substitution hints:    A( x, y, z)    B( x, y, z, w)    F( x, y, z)    G( x, y, z)

Proof of Theorem tfrlem2
StepHypRef Expression
1 abai 771 . . . . 5  |-  ( ( A  e.  On  /\  ( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) )  <-> 
( A  e.  On  /\  ( A  e.  On  ->  ( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) ) )
21albii 1572 . . . 4  |-  ( A. w ( A  e.  On  /\  ( w  e.  A  ->  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) )  <->  A. w ( A  e.  On  /\  ( A  e.  On  ->  (
w  e.  A  -> 
( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) ) )
3 19.28v 1907 . . . 4  |-  ( A. w ( A  e.  On  /\  ( w  e.  A  ->  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) )  <-> 
( A  e.  On  /\ 
A. w ( w  e.  A  ->  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )
4 19.28v 1907 . . . 4  |-  ( A. w ( A  e.  On  /\  ( A  e.  On  ->  (
w  e.  A  -> 
( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )  <->  ( A  e.  On  /\  A. w
( A  e.  On  ->  ( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) ) )
52, 3, 43bitr3ri 268 . . 3  |-  ( ( A  e.  On  /\  A. w ( A  e.  On  ->  ( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `
 w )  =  ( B `  ( G  |`  w ) ) ) ) ) )  <-> 
( A  e.  On  /\ 
A. w ( w  e.  A  ->  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )
6 df-ral 2654 . . . . . 6  |-  ( A. w  e.  A  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) )  <->  A. w
( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) )
76anbi2i 676 . . . . 5  |-  ( ( A  e.  On  /\  A. w  e.  A  ( ( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) )  <->  ( A  e.  On  /\  A. w
( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )
8 fnop 5488 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  <.
x ,  y >.  e.  F )  ->  x  e.  A )
98adantlr 696 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  <. x ,  y >.  e.  F
)  ->  x  e.  A )
10 tfrlem1 6572 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
( F  Fn  A  /\  G  Fn  A
)  ->  ( A. w  e.  A  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) )  ->  A. w  e.  A  ( F `  w )  =  ( G `  w ) ) ) )
1110com12 29 . . . . . . . . . 10  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A  e.  On  ->  ( A. w  e.  A  ( ( F `
 w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) )  ->  A. w  e.  A  ( F `  w )  =  ( G `  w ) ) ) )
1211imp3a 421 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( A  e.  On  /\  A. w  e.  A  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `
 w )  =  ( B `  ( G  |`  w ) ) ) )  ->  A. w  e.  A  ( F `  w )  =  ( G `  w ) ) )
1312adantr 452 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  <. x ,  y >.  e.  F
)  ->  ( ( A  e.  On  /\  A. w  e.  A  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) )  ->  A. w  e.  A  ( F `  w )  =  ( G `  w ) ) )
14 fveq2 5668 . . . . . . . . . 10  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
15 fveq2 5668 . . . . . . . . . 10  |-  ( w  =  x  ->  ( G `  w )  =  ( G `  x ) )
1614, 15eqeq12d 2401 . . . . . . . . 9  |-  ( w  =  x  ->  (
( F `  w
)  =  ( G `
 w )  <->  ( F `  x )  =  ( G `  x ) ) )
1716rspcv 2991 . . . . . . . 8  |-  ( x  e.  A  ->  ( A. w  e.  A  ( F `  w )  =  ( G `  w )  ->  ( F `  x )  =  ( G `  x ) ) )
189, 13, 17sylsyld 54 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  <. x ,  y >.  e.  F
)  ->  ( ( A  e.  On  /\  A. w  e.  A  (
( F `  w
)  =  ( B `
 ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) )  -> 
( F `  x
)  =  ( G `
 x ) ) )
1918imp 419 . . . . . 6  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  <. x ,  y >.  e.  F
)  /\  ( A  e.  On  /\  A. w  e.  A  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `
 w )  =  ( B `  ( G  |`  w ) ) ) ) )  -> 
( F `  x
)  =  ( G `
 x ) )
2019adantlrr 702 . . . . 5  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  /\  ( A  e.  On  /\  A. w  e.  A  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `
 w )  =  ( B `  ( G  |`  w ) ) ) ) )  -> 
( F `  x
)  =  ( G `
 x ) )
217, 20sylan2br 463 . . . 4  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  /\  ( A  e.  On  /\  A. w
( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )  ->  ( F `  x )  =  ( G `  x ) )
22 fnfun 5482 . . . . . . . 8  |-  ( F  Fn  A  ->  Fun  F )
23 fnfun 5482 . . . . . . . 8  |-  ( G  Fn  A  ->  Fun  G )
2422, 23anim12i 550 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( Fun  F  /\  Fun  G ) )
25 funopfv 5705 . . . . . . . . . 10  |-  ( Fun 
F  ->  ( <. x ,  y >.  e.  F  ->  ( F `  x
)  =  y ) )
2625imp 419 . . . . . . . . 9  |-  ( ( Fun  F  /\  <. x ,  y >.  e.  F
)  ->  ( F `  x )  =  y )
27 funopfv 5705 . . . . . . . . . 10  |-  ( Fun 
G  ->  ( <. x ,  z >.  e.  G  ->  ( G `  x
)  =  z ) )
2827imp 419 . . . . . . . . 9  |-  ( ( Fun  G  /\  <. x ,  z >.  e.  G
)  ->  ( G `  x )  =  z )
2926, 28anim12i 550 . . . . . . . 8  |-  ( ( ( Fun  F  /\  <.
x ,  y >.  e.  F )  /\  ( Fun  G  /\  <. x ,  z >.  e.  G
) )  ->  (
( F `  x
)  =  y  /\  ( G `  x )  =  z ) )
3029an4s 800 . . . . . . 7  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  G
) )  ->  (
( F `  x
)  =  y  /\  ( G `  x )  =  z ) )
3124, 30sylan 458 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  ( ( F `
 x )  =  y  /\  ( G `
 x )  =  z ) )
32 eqeq12 2399 . . . . . 6  |-  ( ( ( F `  x
)  =  y  /\  ( G `  x )  =  z )  -> 
( ( F `  x )  =  ( G `  x )  <-> 
y  =  z ) )
3331, 32syl 16 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  ( ( F `
 x )  =  ( G `  x
)  <->  y  =  z ) )
3433adantr 452 . . . 4  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  /\  ( A  e.  On  /\  A. w
( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )  ->  ( ( F `  x )  =  ( G `  x )  <->  y  =  z ) )
3521, 34mpbid 202 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  /\  ( A  e.  On  /\  A. w
( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) )  ->  y  =  z )
365, 35sylan2b 462 . 2  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  /\  ( A  e.  On  /\  A. w
( A  e.  On  ->  ( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) ) ) )  ->  y  =  z )
3736exp43 596 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G )  -> 
( A  e.  On  ->  ( A. w ( A  e.  On  ->  ( w  e.  A  -> 
( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `  w )  =  ( B `  ( G  |`  w ) ) ) ) )  ->  y  =  z ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   A.wral 2649   <.cop 3760   Oncon0 4522    |` cres 4820   Fun wfun 5388    Fn wfn 5389   ` cfv 5394
This theorem is referenced by:  tfrlem5  6577
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-fv 5402
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