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Theorem tfrlem2 6278
Description: Lemma for transfinite recursion. This provides some messy details needed to link tfrlem1 6277 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 773 . . . . 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 1554 . . . 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 2028 . . . 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 2028 . . . 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 269 . . 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 2513 . . . . . 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 678 . . . . 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 5204 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  <.
x ,  y >.  e.  F )  ->  x  e.  A )
98adantlr 698 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  <. x ,  y >.  e.  F
)  ->  x  e.  A )
10 tfrlem1 6277 . . . . . . . . . . 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 422 . . . . . . . . 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 453 . . . . . . . 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 5377 . . . . . . . . . 10  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
15 fveq2 5377 . . . . . . . . . 10  |-  ( w  =  x  ->  ( G `  w )  =  ( G `  x ) )
1614, 15eqeq12d 2267 . . . . . . . . 9  |-  ( w  =  x  ->  (
( F `  w
)  =  ( G `
 w )  <->  ( F `  x )  =  ( G `  x ) ) )
1716rcla4v 2817 . . . . . . . 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 420 . . . . . 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 704 . . . . 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 464 . . . 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 5198 . . . . . . . 8  |-  ( F  Fn  A  ->  Fun  F )
23 fnfun 5198 . . . . . . . 8  |-  ( G  Fn  A  ->  Fun  G )
2422, 23anim12i 551 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( Fun  F  /\  Fun  G ) )
25 funopfv 5414 . . . . . . . . . 10  |-  ( Fun 
F  ->  ( <. x ,  y >.  e.  F  ->  ( F `  x
)  =  y ) )
2625imp 420 . . . . . . . . 9  |-  ( ( Fun  F  /\  <. x ,  y >.  e.  F
)  ->  ( F `  x )  =  y )
27 funopfv 5414 . . . . . . . . . 10  |-  ( Fun 
G  ->  ( <. x ,  z >.  e.  G  ->  ( G `  x
)  =  z ) )
2827imp 420 . . . . . . . . 9  |-  ( ( Fun  G  /\  <. x ,  z >.  e.  G
)  ->  ( G `  x )  =  z )
2926, 28anim12i 551 . . . . . . . 8  |-  ( ( ( Fun  F  /\  <.
x ,  y >.  e.  F )  /\  ( Fun  G  /\  <. x ,  z >.  e.  G
) )  ->  (
( F `  x
)  =  y  /\  ( G `  x )  =  z ) )
3029an4s 802 . . . . . . 7  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  G
) )  ->  (
( F `  x
)  =  y  /\  ( G `  x )  =  z ) )
3124, 30sylan 459 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  ( ( F `
 x )  =  y  /\  ( G `
 x )  =  z ) )
32 eqeq12 2265 . . . . . 6  |-  ( ( ( F `  x
)  =  y  /\  ( G `  x )  =  z )  -> 
( ( F `  x )  =  ( G `  x )  <-> 
y  =  z ) )
3331, 32syl 17 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  G ) )  ->  ( ( F `
 x )  =  ( G `  x
)  <->  y  =  z ) )
3433adantr 453 . . . 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 203 . . 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 463 . 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 598 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 6    <-> wb 178    /\ wa 360   A.wal 1532    = wceq 1619    e. wcel 1621   A.wral 2509   <.cop 3547   Oncon0 4285    |` cres 4582   Fun wfun 4586    Fn wfn 4587   ` cfv 4592
This theorem is referenced by:  tfrlem5  6282
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-fv 4608
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