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

Theorem fvmptdf 5977
Description: Alternate deduction version of fvmpt 5964, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
fvmptdf.1  |-  ( ph  ->  A  e.  D )
fvmptdf.2  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
fvmptdf.3  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  B  ->  ps ) )
fvmptdf.4  |-  F/_ x F
fvmptdf.5  |-  F/ x ps
Assertion
Ref Expression
fvmptdf  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps )
)
Distinct variable groups:    x, A    x, D    ph, x
Allowed substitution hints:    ps( x)    B( x)    F( x)    V( x)

Proof of Theorem fvmptdf
StepHypRef Expression
1 nfv 1754 . 2  |-  F/ x ph
2 fvmptdf.4 . . . 4  |-  F/_ x F
3 nfmpt1 4515 . . . 4  |-  F/_ x
( x  e.  D  |->  B )
42, 3nfeq 2602 . . 3  |-  F/ x  F  =  ( x  e.  D  |->  B )
5 fvmptdf.5 . . 3  |-  F/ x ps
64, 5nfim 1978 . 2  |-  F/ x
( F  =  ( x  e.  D  |->  B )  ->  ps )
7 fvmptdf.1 . . . 4  |-  ( ph  ->  A  e.  D )
8 elex 3096 . . . 4  |-  ( A  e.  D  ->  A  e.  _V )
97, 8syl 17 . . 3  |-  ( ph  ->  A  e.  _V )
10 isset 3091 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
119, 10sylib 199 . 2  |-  ( ph  ->  E. x  x  =  A )
12 fveq1 5880 . . 3  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A ) )
13 simpr 462 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
1413fveq2d 5885 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  x
)  =  ( ( x  e.  D  |->  B ) `  A ) )
157adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  =  A )  ->  A  e.  D )
1613, 15eqeltrd 2517 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  e.  D )
17 fvmptdf.2 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
18 eqid 2429 . . . . . . . 8  |-  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B )
1918fvmpt2 5973 . . . . . . 7  |-  ( ( x  e.  D  /\  B  e.  V )  ->  ( ( x  e.  D  |->  B ) `  x )  =  B )
2016, 17, 19syl2anc 665 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  x
)  =  B )
2114, 20eqtr3d 2472 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  D  |->  B ) `  A
)  =  B )
2221eqeq2d 2443 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A )  <-> 
( F `  A
)  =  B ) )
23 fvmptdf.3 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  B  ->  ps ) )
2422, 23sylbid 218 . . 3  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A )  ->  ps ) )
2512, 24syl5 33 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps ) )
261, 6, 11, 25exlimdd 2037 1  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1659   F/wnf 1663    e. wcel 1870   F/_wnfc 2577   _Vcvv 3087    |-> cmpt 4484   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fv 5609
This theorem is referenced by:  fvmptdv  5978  yonedalem4b  16112
  Copyright terms: Public domain W3C validator