ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fvmptt Unicode version

Theorem fvmptt 5378
Description: Closed theorem form of fvmpt 5365. (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 944 . . 3  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  F  =  ( x  e.  D  |->  B )  /\  ( A  e.  D  /\  C  e.  V
) )  ->  F  =  ( x  e.  D  |->  B ) )
21fveq1d 5291 . 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 2406 . . . . 5  |-  ( A  e.  D  <->  E. x  e.  D  x  =  A )
4 elex 2630 . . . . . 6  |-  ( C  e.  V  ->  C  e.  _V )
5 nfa1 1479 . . . . . . 7  |-  F/ x A. x ( x  =  A  ->  B  =  C )
6 nfv 1466 . . . . . . . 8  |-  F/ x  C  e.  _V
7 nffvmpt1 5300 . . . . . . . . 9  |-  F/_ x
( ( x  e.  D  |->  B ) `  A )
87nfeq1 2238 . . . . . . . 8  |-  F/ x
( ( x  e.  D  |->  B ) `  A )  =  C
96, 8nfim 1509 . . . . . . 7  |-  F/ x
( C  e.  _V  ->  ( ( x  e.  D  |->  B ) `  A )  =  C )
10 simprl 498 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  x  e.  D )
11 simplr 497 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  B  =  C )
12 simprr 499 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  C  e.  _V )
1311, 12eqeltrd 2164 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  B  e.  _V )
14 eqid 2088 . . . . . . . . . . . . . 14  |-  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B )
1514fvmpt2 5370 . . . . . . . . . . . . 13  |-  ( ( x  e.  D  /\  B  e.  _V )  ->  ( ( x  e.  D  |->  B ) `  x )  =  B )
1610, 13, 15syl2anc 403 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  x )  =  B )
17 simpll 496 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  ->  x  =  A )
1817fveq2d 5293 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  x )  =  ( ( x  e.  D  |->  B ) `  A
) )
1916, 18, 113eqtr3d 2128 . . . . . . . . . . 11  |-  ( ( ( x  =  A  /\  B  =  C )  /\  ( x  e.  D  /\  C  e.  _V ) )  -> 
( ( x  e.  D  |->  B ) `  A )  =  C )
2019exp43 364 . . . . . . . . . 10  |-  ( x  =  A  ->  ( B  =  C  ->  ( x  e.  D  -> 
( C  e.  _V  ->  ( ( x  e.  D  |->  B ) `  A )  =  C ) ) ) )
2120a2i 11 . . . . . . . . 9  |-  ( ( x  =  A  ->  B  =  C )  ->  ( x  =  A  ->  ( x  e.  D  ->  ( C  e.  _V  ->  ( (
x  e.  D  |->  B ) `  A )  =  C ) ) ) )
2221com23 77 . . . . . . . 8  |-  ( ( x  =  A  ->  B  =  C )  ->  ( x  e.  D  ->  ( x  =  A  ->  ( C  e. 
_V  ->  ( ( x  e.  D  |->  B ) `
 A )  =  C ) ) ) )
2322sps 1475 . . . . . . 7  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  (
x  e.  D  -> 
( x  =  A  ->  ( C  e. 
_V  ->  ( ( x  e.  D  |->  B ) `
 A )  =  C ) ) ) )
245, 9, 23rexlimd 2486 . . . . . 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 150 . . . 4  |-  ( A. x ( x  =  A  ->  B  =  C )  ->  ( A  e.  D  ->  ( C  e.  V  -> 
( ( x  e.  D  |->  B ) `  A )  =  C ) ) )
2726imp32 253 . . 3  |-  ( ( A. x ( x  =  A  ->  B  =  C )  /\  ( A  e.  D  /\  C  e.  V )
)  ->  ( (
x  e.  D  |->  B ) `  A )  =  C )
28273adant2 962 . 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 2120 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 102    /\ w3a 924   A.wal 1287    = wceq 1289    e. wcel 1438   E.wrex 2360   _Vcvv 2619    |-> cmpt 3891   ` cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator