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Theorem fvmptt 5849
Description: Closed theorem form of fvmpt 5835. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
Assertion
Ref Expression
fvmptt  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  F  =  ( x  e.  D  |->  B )  /\  ( A  e.  D  /\  C  e.  V
) )  ->  ( F `  A )  =  C )
Distinct variable groups:    x, A    x, C    x, D
Allowed substitution hints:    B( x)    F( x)    V( x)

Proof of Theorem fvmptt
StepHypRef Expression
1 simp2 959 . . 3  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  F  =  ( x  e.  D  |->  B )  /\  ( A  e.  D  /\  C  e.  V
) )  ->  F  =  ( x  e.  D  |->  B ) )
21fveq1d 5759 . 2  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  F  =  ( x  e.  D  |->  B )  /\  ( A  e.  D  /\  C  e.  V
) )  ->  ( F `  A )  =  ( ( x  e.  D  |->  B ) `
 A ) )
3 risset 2759 . . . . 5  |-  ( A  e.  D  <->  E. x  e.  D  x  =  A )
4 elex 2970 . . . . . 6  |-  ( C  e.  V  ->  C  e.  _V )
5 nfa1 1808 . . . . . . 7  |-  F/ x A. x ( x  =  A  ->  B  =  C )
6 nfv 1630 . . . . . . . 8  |-  F/ x  C  e.  _V
7 nffvmpt1 5765 . . . . . . . . 9  |-  F/_ x
( ( x  e.  D  |->  B ) `  A )
87nfeq1 2587 . . . . . . . 8  |-  F/ x
( ( x  e.  D  |->  B ) `  A )  =  C
96, 8nfim 1834 . . . . . . 7  |-  F/ x
( C  e.  _V  ->  ( ( x  e.  D  |->  B ) `  A )  =  C )
10 simprl 734 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  x  e.  D )
11 simplr 733 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  B  =  C )
12 simprr 735 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  C  e.  _V )
1311, 12eqeltrd 2516 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  B  e.  _V )
14 eqid 2442 . . . . . . . . . . . . . 14  |-  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B )
1514fvmpt2 5841 . . . . . . . . . . . . 13  |-  ( ( x  e.  D  /\  B  e.  _V )  ->  ( ( x  e.  D  |->  B ) `  x )  =  B )
1610, 13, 15syl2anc 644 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  x )  =  B )
17 simpll 732 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  x  =  A )
1817fveq2d 5761 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  x )  =  ( ( x  e.  D  |->  B ) `  A
) )
1916, 18, 113eqtr3d 2482 . . . . . . . . . . 11  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  A )  =  C )
2019exp43 597 . . . . . . . . . 10  |-  ( x  =  A  ->  ( B  =  C  ->  ( x  e.  D  -> 
( C  e.  _V  ->  ( ( x  e.  D  |->  B ) `  A )  =  C ) ) ) )
2120a2i 13 . . . . . . . . 9  |-  ( ( x  =  A  ->  B  =  C )  ->  ( x  =  A  ->  ( x  e.  D  ->  ( C  e.  _V  ->  ( (
x  e.  D  |->  B ) `  A )  =  C ) ) ) )
2221com23 75 . . . . . . . 8  |-  ( ( x  =  A  ->  B  =  C )  ->  ( x  e.  D  ->  ( x  =  A  ->  ( C  e. 
_V  ->  ( ( x  e.  D  |->  B ) `
 A )  =  C ) ) ) )
2322sps 1772 . . . . . . 7  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  (
x  e.  D  -> 
( x  =  A  ->  ( C  e. 
_V  ->  ( ( x  e.  D  |->  B ) `
 A )  =  C ) ) ) )
245, 9, 23rexlimd 2833 . . . . . 6  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  ( E. x  e.  D  x  =  A  ->  ( C  e.  _V  ->  ( ( x  e.  D  |->  B ) `  A
)  =  C ) ) )
254, 24syl7 66 . . . . 5  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  ( E. x  e.  D  x  =  A  ->  ( C  e.  V  -> 
( ( x  e.  D  |->  B ) `  A )  =  C ) ) )
263, 25syl5bi 210 . . . 4  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  ( A  e.  D  ->  ( C  e.  V  -> 
( ( x  e.  D  |->  B ) `  A )  =  C ) ) )
2726imp32 424 . . 3  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  ( A  e.  D  /\  C  e.  V )
)  ->  ( (
x  e.  D  |->  B ) `  A )  =  C )
28273adant2 977 . 2  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  F  =  ( x  e.  D  |->  B )  /\  ( A  e.  D  /\  C  e.  V
) )  ->  (
( x  e.  D  |->  B ) `  A
)  =  C )
292, 28eqtrd 2474 1  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  F  =  ( x  e.  D  |->  B )  /\  ( A  e.  D  /\  C  e.  V
) )  ->  ( F `  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   A.wal 1550    = wceq 1653    e. wcel 1727   E.wrex 2712   _Vcvv 2962    e. cmpt 4291   ` cfv 5483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fv 5491
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