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Theorem bnj1385 32178
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 1674 . . . . . . 7  |-  F/ h
( f  e.  A  ->  Fun  f )
2 bnj1385.4 . . . . . . . . . 10  |-  ( x  e.  A  ->  A. f  x  e.  A )
32nfcii 2606 . . . . . . . . 9  |-  F/_ f A
43nfcri 2609 . . . . . . . 8  |-  F/ f  h  e.  A
5 nfv 1674 . . . . . . . 8  |-  F/ f Fun  h
64, 5nfim 1858 . . . . . . 7  |-  F/ f ( h  e.  A  ->  Fun  h )
7 eleq1 2526 . . . . . . . 8  |-  ( f  =  h  ->  (
f  e.  A  <->  h  e.  A ) )
8 funeq 5548 . . . . . . . 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 1981 . . . . . 6  |-  ( A. f ( f  e.  A  ->  Fun  f )  <->  A. h ( h  e.  A  ->  Fun  h ) )
11 df-ral 2804 . . . . . 6  |-  ( A. f  e.  A  Fun  f 
<-> 
A. f ( f  e.  A  ->  Fun  f ) )
12 df-ral 2804 . . . . . 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 1674 . . . . . 6  |-  F/ h
( f  e.  A  ->  A. g  e.  A  ( f  |`  D )  =  ( g  |`  D ) )
18 nfv 1674 . . . . . . . 8  |-  F/ f ( h  |`  E )  =  ( g  |`  E )
193, 18nfral 2888 . . . . . . 7  |-  F/ f A. g  e.  A  ( h  |`  E )  =  ( g  |`  E )
204, 19nfim 1858 . . . . . 6  |-  F/ f ( h  e.  A  ->  A. g  e.  A  ( h  |`  E )  =  ( g  |`  E ) )
21 dmeq 5151 . . . . . . . . . . . . 13  |-  ( f  =  h  ->  dom  f  =  dom  h )
2221ineq1d 3662 . . . . . . . . . . . 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 2520 . . . . . . . . . . 11  |-  ( f  =  h  ->  D  =  E )
2625reseq2d 5221 . . . . . . . . . 10  |-  ( f  =  h  ->  (
f  |`  D )  =  ( f  |`  E ) )
27 reseq1 5215 . . . . . . . . . 10  |-  ( f  =  h  ->  (
f  |`  E )  =  ( h  |`  E ) )
2826, 27eqtrd 2495 . . . . . . . . 9  |-  ( f  =  h  ->  (
f  |`  D )  =  ( h  |`  E ) )
2925reseq2d 5221 . . . . . . . . 9  |-  ( f  =  h  ->  (
g  |`  D )  =  ( g  |`  E ) )
3028, 29eqeq12d 2476 . . . . . . . 8  |-  ( f  =  h  ->  (
( f  |`  D )  =  ( g  |`  D )  <->  ( h  |`  E )  =  ( g  |`  E )
) )
3130ralbidv 2846 . . . . . . 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 1981 . . . . 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 2804 . . . . 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 2804 . . . . 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 32177 . 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 1368    = wceq 1370    e. wcel 1758   A.wral 2799    i^i cin 3438   U.cuni 4202   dom cdm 4951    |` cres 4953   Fun wfun 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-res 4963  df-iota 5492  df-fun 5531  df-fv 5537
This theorem is referenced by:  bnj1386  32179
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