MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmptt Unicode version

Theorem fvmptt 5722
Description: Closed theorem form of fvmpt 5709. (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 957 . . 3  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  F  =  ( x  e.  D  |->  B )  /\  ( A  e.  D  /\  C  e.  V
) )  ->  F  =  ( x  e.  D  |->  B ) )
21fveq1d 5634 . 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 2675 . . . . 5  |-  ( A  e.  D  <->  E. x  e.  D  x  =  A )
4 elex 2881 . . . . . 6  |-  ( C  e.  V  ->  C  e.  _V )
5 nfa1 1794 . . . . . . 7  |-  F/ x A. x ( x  =  A  ->  B  =  C )
6 nfv 1624 . . . . . . . 8  |-  F/ x  C  e.  _V
7 nfmpt1 4211 . . . . . . . . . 10  |-  F/_ x
( x  e.  D  |->  B )
8 nfcv 2502 . . . . . . . . . 10  |-  F/_ x A
97, 8nffv 5639 . . . . . . . . 9  |-  F/_ x
( ( x  e.  D  |->  B ) `  A )
109nfeq1 2511 . . . . . . . 8  |-  F/ x
( ( x  e.  D  |->  B ) `  A )  =  C
116, 10nfim 1820 . . . . . . 7  |-  F/ x
( C  e.  _V  ->  ( ( x  e.  D  |->  B ) `  A )  =  C )
12 simprl 732 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  x  e.  D )
13 simplr 731 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  B  =  C )
14 simprr 733 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  C  e.  _V )
1513, 14eqeltrd 2440 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  B  e.  _V )
16 eqid 2366 . . . . . . . . . . . . . 14  |-  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B )
1716fvmpt2 5715 . . . . . . . . . . . . 13  |-  ( ( x  e.  D  /\  B  e.  _V )  ->  ( ( x  e.  D  |->  B ) `  x )  =  B )
1812, 15, 17syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  x )  =  B )
19 simpll 730 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  x  =  A )
2019fveq2d 5636 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  x )  =  ( ( x  e.  D  |->  B ) `  A
) )
2118, 20, 133eqtr3d 2406 . . . . . . . . . . 11  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  A )  =  C )
2221exp43 595 . . . . . . . . . 10  |-  ( x  =  A  ->  ( B  =  C  ->  ( x  e.  D  -> 
( C  e.  _V  ->  ( ( x  e.  D  |->  B ) `  A )  =  C ) ) ) )
2322a2i 12 . . . . . . . . 9  |-  ( ( x  =  A  ->  B  =  C )  ->  ( x  =  A  ->  ( x  e.  D  ->  ( C  e.  _V  ->  ( (
x  e.  D  |->  B ) `  A )  =  C ) ) ) )
2423com23 72 . . . . . . . 8  |-  ( ( x  =  A  ->  B  =  C )  ->  ( x  e.  D  ->  ( x  =  A  ->  ( C  e. 
_V  ->  ( ( x  e.  D  |->  B ) `
 A )  =  C ) ) ) )
2524sps 1760 . . . . . . 7  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  (
x  e.  D  -> 
( x  =  A  ->  ( C  e. 
_V  ->  ( ( x  e.  D  |->  B ) `
 A )  =  C ) ) ) )
265, 11, 25rexlimd 2749 . . . . . 6  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  ( E. x  e.  D  x  =  A  ->  ( C  e.  _V  ->  ( ( x  e.  D  |->  B ) `  A
)  =  C ) ) )
274, 26syl7 63 . . . . 5  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  ( E. x  e.  D  x  =  A  ->  ( C  e.  V  -> 
( ( x  e.  D  |->  B ) `  A )  =  C ) ) )
283, 27syl5bi 208 . . . 4  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  ( A  e.  D  ->  ( C  e.  V  -> 
( ( x  e.  D  |->  B ) `  A )  =  C ) ) )
2928imp32 422 . . 3  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  ( A  e.  D  /\  C  e.  V )
)  ->  ( (
x  e.  D  |->  B ) `  A )  =  C )
30293adant2 975 . 2  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  F  =  ( x  e.  D  |->  B )  /\  ( A  e.  D  /\  C  e.  V
) )  ->  (
( x  e.  D  |->  B ) `  A
)  =  C )
312, 30eqtrd 2398 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 358    /\ w3a 935   A.wal 1545    = wceq 1647    e. wcel 1715   E.wrex 2629   _Vcvv 2873    e. cmpt 4179   ` cfv 5358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fv 5366
  Copyright terms: Public domain W3C validator