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

Theorem fvmptt 5812
Description: Closed theorem form of fvmpt 5798. (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 958 . . 3  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  F  =  ( x  e.  D  |->  B )  /\  ( A  e.  D  /\  C  e.  V
) )  ->  F  =  ( x  e.  D  |->  B ) )
21fveq1d 5722 . 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 2745 . . . . 5  |-  ( A  e.  D  <->  E. x  e.  D  x  =  A )
4 elex 2956 . . . . . 6  |-  ( C  e.  V  ->  C  e.  _V )
5 nfa1 1806 . . . . . . 7  |-  F/ x A. x ( x  =  A  ->  B  =  C )
6 nfv 1629 . . . . . . . 8  |-  F/ x  C  e.  _V
7 nffvmpt1 5728 . . . . . . . . 9  |-  F/_ x
( ( x  e.  D  |->  B ) `  A )
87nfeq1 2580 . . . . . . . 8  |-  F/ x
( ( x  e.  D  |->  B ) `  A )  =  C
96, 8nfim 1832 . . . . . . 7  |-  F/ x
( C  e.  _V  ->  ( ( x  e.  D  |->  B ) `  A )  =  C )
10 simprl 733 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  x  e.  D )
11 simplr 732 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  B  =  C )
12 simprr 734 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  C  e.  _V )
1311, 12eqeltrd 2509 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  B  e.  _V )
14 eqid 2435 . . . . . . . . . . . . . 14  |-  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B )
1514fvmpt2 5804 . . . . . . . . . . . . 13  |-  ( ( x  e.  D  /\  B  e.  _V )  ->  ( ( x  e.  D  |->  B ) `  x )  =  B )
1610, 13, 15syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  x )  =  B )
17 simpll 731 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  x  =  A )
1817fveq2d 5724 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  x )  =  ( ( x  e.  D  |->  B ) `  A
) )
1916, 18, 113eqtr3d 2475 . . . . . . . . . . 11  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  A )  =  C )
2019exp43 596 . . . . . . . . . 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 74 . . . . . . . 8  |-  ( ( x  =  A  ->  B  =  C )  ->  ( x  e.  D  ->  ( x  =  A  ->  ( C  e. 
_V  ->  ( ( x  e.  D  |->  B ) `
 A )  =  C ) ) ) )
2322sps 1770 . . . . . . 7  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  (
x  e.  D  -> 
( x  =  A  ->  ( C  e. 
_V  ->  ( ( x  e.  D  |->  B ) `
 A )  =  C ) ) ) )
245, 9, 23rexlimd 2819 . . . . . 6  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  ( E. x  e.  D  x  =  A  ->  ( C  e.  _V  ->  ( ( x  e.  D  |->  B ) `  A
)  =  C ) ) )
254, 24syl7 65 . . . . 5  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  ( E. x  e.  D  x  =  A  ->  ( C  e.  V  -> 
( ( x  e.  D  |->  B ) `  A )  =  C ) ) )
263, 25syl5bi 209 . . . 4  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  ( A  e.  D  ->  ( C  e.  V  -> 
( ( x  e.  D  |->  B ) `  A )  =  C ) ) )
2726imp32 423 . . 3  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  ( A  e.  D  /\  C  e.  V )
)  ->  ( (
x  e.  D  |->  B ) `  A )  =  C )
28273adant2 976 . 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 2467 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 359    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948    e. cmpt 4258   ` cfv 5446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454
  Copyright terms: Public domain W3C validator