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Theorem fvmptt 5963
Description: Closed theorem form of fvmpt 5948. (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 997 . . 3  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  F  =  ( x  e.  D  |->  B )  /\  ( A  e.  D  /\  C  e.  V
) )  ->  F  =  ( x  e.  D  |->  B ) )
21fveq1d 5866 . 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 2987 . . . . 5  |-  ( A  e.  D  <->  E. x  e.  D  x  =  A )
4 elex 3122 . . . . . 6  |-  ( C  e.  V  ->  C  e.  _V )
5 nfa1 1845 . . . . . . 7  |-  F/ x A. x ( x  =  A  ->  B  =  C )
6 nfv 1683 . . . . . . . 8  |-  F/ x  C  e.  _V
7 nffvmpt1 5872 . . . . . . . . 9  |-  F/_ x
( ( x  e.  D  |->  B ) `  A )
87nfeq1 2644 . . . . . . . 8  |-  F/ x
( ( x  e.  D  |->  B ) `  A )  =  C
96, 8nfim 1867 . . . . . . 7  |-  F/ x
( C  e.  _V  ->  ( ( x  e.  D  |->  B ) `  A )  =  C )
10 simprl 755 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  x  e.  D )
11 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  B  =  C )
12 simprr 756 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  C  e.  _V )
1311, 12eqeltrd 2555 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  B  e.  _V )
14 eqid 2467 . . . . . . . . . . . . . 14  |-  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B )
1514fvmpt2 5955 . . . . . . . . . . . . 13  |-  ( ( x  e.  D  /\  B  e.  _V )  ->  ( ( x  e.  D  |->  B ) `  x )  =  B )
1610, 13, 15syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  x )  =  B )
17 simpll 753 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  x  =  A )
1817fveq2d 5868 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  x )  =  ( ( x  e.  D  |->  B ) `  A
) )
1916, 18, 113eqtr3d 2516 . . . . . . . . . . 11  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  A )  =  C )
2019exp43 612 . . . . . . . . . 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 78 . . . . . . . 8  |-  ( ( x  =  A  ->  B  =  C )  ->  ( x  e.  D  ->  ( x  =  A  ->  ( C  e. 
_V  ->  ( ( x  e.  D  |->  B ) `
 A )  =  C ) ) ) )
2322sps 1814 . . . . . . 7  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  (
x  e.  D  -> 
( x  =  A  ->  ( C  e. 
_V  ->  ( ( x  e.  D  |->  B ) `
 A )  =  C ) ) ) )
245, 9, 23rexlimd 2947 . . . . . 6  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  ( E. x  e.  D  x  =  A  ->  ( C  e.  _V  ->  ( ( x  e.  D  |->  B ) `  A
)  =  C ) ) )
254, 24syl7 68 . . . . 5  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  ( E. x  e.  D  x  =  A  ->  ( C  e.  V  -> 
( ( x  e.  D  |->  B ) `  A )  =  C ) ) )
263, 25syl5bi 217 . . . 4  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  ( A  e.  D  ->  ( C  e.  V  -> 
( ( x  e.  D  |->  B ) `  A )  =  C ) ) )
2726imp32 433 . . 3  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  ( A  e.  D  /\  C  e.  V )
)  ->  ( (
x  e.  D  |->  B ) `  A )  =  C )
28273adant2 1015 . 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 2508 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113    |-> cmpt 4505   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fv 5594
This theorem is referenced by: (None)
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