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Theorem bnj1385 31713
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1385.1  |-  ( ph  <->  A. f  e.  A  Fun  f )
bnj1385.2  |-  D  =  ( dom  f  i^i 
dom  g )
bnj1385.3  |-  ( ps  <->  (
ph  /\  A. f  e.  A  A. g  e.  A  ( f  |`  D )  =  ( g  |`  D )
) )
bnj1385.4  |-  ( x  e.  A  ->  A. f  x  e.  A )
bnj1385.5  |-  ( ph'  <->  A. h  e.  A  Fun  h )
bnj1385.6  |-  E  =  ( dom  h  i^i 
dom  g )
bnj1385.7  |-  ( ps'  <->  ( ph' 
/\  A. h  e.  A  A. g  e.  A  ( h  |`  E )  =  ( g  |`  E ) ) )
Assertion
Ref Expression
bnj1385  |-  ( ps 
->  Fun  U. A )
Distinct variable groups:    A, g, h, x    D, h    f, E    f, g, h, x   
g, ph'
Allowed substitution hints:    ph( x, f, g, h)    ps( x, f, g, h)    A( f)    D( x, f, g)    E( x, g, h)    ph'( x, f, h)    ps'( x, f, g, h)

Proof of Theorem bnj1385
StepHypRef Expression
1 nfv 1673 . . . . . . 7  |-  F/ h
( f  e.  A  ->  Fun  f )
2 bnj1385.4 . . . . . . . . . 10  |-  ( x  e.  A  ->  A. f  x  e.  A )
32nfcii 2565 . . . . . . . . 9  |-  F/_ f A
43nfcri 2568 . . . . . . . 8  |-  F/ f  h  e.  A
5 nfv 1673 . . . . . . . 8  |-  F/ f Fun  h
64, 5nfim 1852 . . . . . . 7  |-  F/ f ( h  e.  A  ->  Fun  h )
7 eleq1 2498 . . . . . . . 8  |-  ( f  =  h  ->  (
f  e.  A  <->  h  e.  A ) )
8 funeq 5432 . . . . . . . 8  |-  ( f  =  h  ->  ( Fun  f  <->  Fun  h ) )
97, 8imbi12d 320 . . . . . . 7  |-  ( f  =  h  ->  (
( f  e.  A  ->  Fun  f )  <->  ( h  e.  A  ->  Fun  h
) ) )
101, 6, 9cbval 1969 . . . . . 6  |-  ( A. f ( f  e.  A  ->  Fun  f )  <->  A. h ( h  e.  A  ->  Fun  h ) )
11 df-ral 2715 . . . . . 6  |-  ( A. f  e.  A  Fun  f 
<-> 
A. f ( f  e.  A  ->  Fun  f ) )
12 df-ral 2715 . . . . . 6  |-  ( A. h  e.  A  Fun  h 
<-> 
A. h ( h  e.  A  ->  Fun  h ) )
1310, 11, 123bitr4i 277 . . . . 5  |-  ( A. f  e.  A  Fun  f 
<-> 
A. h  e.  A  Fun  h )
14 bnj1385.1 . . . . 5  |-  ( ph  <->  A. f  e.  A  Fun  f )
15 bnj1385.5 . . . . 5  |-  ( ph'  <->  A. h  e.  A  Fun  h )
1613, 14, 153bitr4i 277 . . . 4  |-  ( ph  <->  ph' )
17 nfv 1673 . . . . . 6  |-  F/ h
( f  e.  A  ->  A. g  e.  A  ( f  |`  D )  =  ( g  |`  D ) )
18 nfv 1673 . . . . . . . 8  |-  F/ f ( h  |`  E )  =  ( g  |`  E )
193, 18nfral 2764 . . . . . . 7  |-  F/ f A. g  e.  A  ( h  |`  E )  =  ( g  |`  E )
204, 19nfim 1852 . . . . . 6  |-  F/ f ( h  e.  A  ->  A. g  e.  A  ( h  |`  E )  =  ( g  |`  E ) )
21 dmeq 5035 . . . . . . . . . . . . 13  |-  ( f  =  h  ->  dom  f  =  dom  h )
2221ineq1d 3546 . . . . . . . . . . . 12  |-  ( f  =  h  ->  ( dom  f  i^i  dom  g
)  =  ( dom  h  i^i  dom  g
) )
23 bnj1385.2 . . . . . . . . . . . 12  |-  D  =  ( dom  f  i^i 
dom  g )
24 bnj1385.6 . . . . . . . . . . . 12  |-  E  =  ( dom  h  i^i 
dom  g )
2522, 23, 243eqtr4g 2495 . . . . . . . . . . 11  |-  ( f  =  h  ->  D  =  E )
2625reseq2d 5105 . . . . . . . . . 10  |-  ( f  =  h  ->  (
f  |`  D )  =  ( f  |`  E ) )
27 reseq1 5099 . . . . . . . . . 10  |-  ( f  =  h  ->  (
f  |`  E )  =  ( h  |`  E ) )
2826, 27eqtrd 2470 . . . . . . . . 9  |-  ( f  =  h  ->  (
f  |`  D )  =  ( h  |`  E ) )
2925reseq2d 5105 . . . . . . . . 9  |-  ( f  =  h  ->  (
g  |`  D )  =  ( g  |`  E ) )
3028, 29eqeq12d 2452 . . . . . . . 8  |-  ( f  =  h  ->  (
( f  |`  D )  =  ( g  |`  D )  <->  ( h  |`  E )  =  ( g  |`  E )
) )
3130ralbidv 2730 . . . . . . 7  |-  ( f  =  h  ->  ( A. g  e.  A  ( f  |`  D )  =  ( g  |`  D )  <->  A. g  e.  A  ( h  |`  E )  =  ( g  |`  E )
) )
327, 31imbi12d 320 . . . . . 6  |-  ( f  =  h  ->  (
( f  e.  A  ->  A. g  e.  A  ( f  |`  D )  =  ( g  |`  D ) )  <->  ( h  e.  A  ->  A. g  e.  A  ( h  |`  E )  =  ( g  |`  E )
) ) )
3317, 20, 32cbval 1969 . . . . 5  |-  ( A. f ( f  e.  A  ->  A. g  e.  A  ( f  |`  D )  =  ( g  |`  D )
)  <->  A. h ( h  e.  A  ->  A. g  e.  A  ( h  |`  E )  =  ( g  |`  E )
) )
34 df-ral 2715 . . . . 5  |-  ( A. f  e.  A  A. g  e.  A  (
f  |`  D )  =  ( g  |`  D )  <->  A. f ( f  e.  A  ->  A. g  e.  A  ( f  |`  D )  =  ( g  |`  D )
) )
35 df-ral 2715 . . . . 5  |-  ( A. h  e.  A  A. g  e.  A  (
h  |`  E )  =  ( g  |`  E )  <->  A. h ( h  e.  A  ->  A. g  e.  A  ( h  |`  E )  =  ( g  |`  E )
) )
3633, 34, 353bitr4i 277 . . . 4  |-  ( A. f  e.  A  A. g  e.  A  (
f  |`  D )  =  ( g  |`  D )  <->  A. h  e.  A  A. g  e.  A  ( h  |`  E )  =  ( g  |`  E ) )
3716, 36anbi12i 697 . . 3  |-  ( (
ph  /\  A. f  e.  A  A. g  e.  A  ( f  |`  D )  =  ( g  |`  D )
)  <->  ( ph'  /\  A. h  e.  A  A. g  e.  A  (
h  |`  E )  =  ( g  |`  E ) ) )
38 bnj1385.3 . . 3  |-  ( ps  <->  (
ph  /\  A. f  e.  A  A. g  e.  A  ( f  |`  D )  =  ( g  |`  D )
) )
39 bnj1385.7 . . 3  |-  ( ps'  <->  ( ph' 
/\  A. h  e.  A  A. g  e.  A  ( h  |`  E )  =  ( g  |`  E ) ) )
4037, 38, 393bitr4i 277 . 2  |-  ( ps  <->  ps' )
4115, 24, 39bnj1383 31712 . 2  |-  ( ps'  ->  Fun  U. A )
4240, 41sylbi 195 1  |-  ( ps 
->  Fun  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369    e. wcel 1756   A.wral 2710    i^i cin 3322   U.cuni 4086   dom cdm 4835    |` cres 4837   Fun wfun 5407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-res 4847  df-iota 5376  df-fun 5415  df-fv 5421
This theorem is referenced by:  bnj1386  31714
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