ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ofrfval2 Unicode version

Theorem ofrfval2 6074
Description: The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval2.1  |-  ( ph  ->  A  e.  V )
offval2.2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  W )
offval2.3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  X )
offval2.4  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
offval2.5  |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )
Assertion
Ref Expression
ofrfval2  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  A  B R C ) )
Distinct variable groups:    x, A    ph, x    x, R
Allowed substitution hints:    B( x)    C( x)    F( x)    G( x)    V( x)    W( x)    X( x)

Proof of Theorem ofrfval2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 offval2.2 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  W )
21ralrimiva 2543 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  W )
3 eqid 2170 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5322 . . . . 5  |-  ( A. x  e.  A  B  e.  W  ->  ( x  e.  A  |->  B )  Fn  A )
52, 4syl 14 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B )  Fn  A
)
6 offval2.4 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
76fneq1d 5286 . . . 4  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 166 . . 3  |-  ( ph  ->  F  Fn  A )
9 offval2.3 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  X )
109ralrimiva 2543 . . . . 5  |-  ( ph  ->  A. x  e.  A  C  e.  X )
11 eqid 2170 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1211fnmpt 5322 . . . . 5  |-  ( A. x  e.  A  C  e.  X  ->  ( x  e.  A  |->  C )  Fn  A )
1310, 12syl 14 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  C )  Fn  A
)
14 offval2.5 . . . . 5  |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )
1514fneq1d 5286 . . . 4  |-  ( ph  ->  ( G  Fn  A  <->  ( x  e.  A  |->  C )  Fn  A ) )
1613, 15mpbird 166 . . 3  |-  ( ph  ->  G  Fn  A )
17 offval2.1 . . 3  |-  ( ph  ->  A  e.  V )
18 inidm 3336 . . 3  |-  ( A  i^i  A )  =  A
196adantr 274 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  F  =  ( x  e.  A  |->  B ) )
2019fveq1d 5496 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( ( x  e.  A  |->  B ) `
 y ) )
2114adantr 274 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  G  =  ( x  e.  A  |->  C ) )
2221fveq1d 5496 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( G `  y )  =  ( ( x  e.  A  |->  C ) `
 y ) )
238, 16, 17, 17, 18, 20, 22ofrfval 6066 . 2  |-  ( ph  ->  ( F  oR R G  <->  A. y  e.  A  ( (
x  e.  A  |->  B ) `  y ) R ( ( x  e.  A  |->  C ) `
 y ) ) )
24 nffvmpt1 5505 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  B ) `  y )
25 nfcv 2312 . . . . 5  |-  F/_ x R
26 nffvmpt1 5505 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  C ) `  y )
2724, 25, 26nfbr 4033 . . . 4  |-  F/ x
( ( x  e.  A  |->  B ) `  y ) R ( ( x  e.  A  |->  C ) `  y
)
28 nfv 1521 . . . 4  |-  F/ y ( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
)
29 fveq2 5494 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  B ) `  y
)  =  ( ( x  e.  A  |->  B ) `  x ) )
30 fveq2 5494 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  C ) `  y
)  =  ( ( x  e.  A  |->  C ) `  x ) )
3129, 30breq12d 4000 . . . 4  |-  ( y  =  x  ->  (
( ( x  e.  A  |->  B ) `  y ) R ( ( x  e.  A  |->  C ) `  y
)  <->  ( ( x  e.  A  |->  B ) `
 x ) R ( ( x  e.  A  |->  C ) `  x ) ) )
3227, 28, 31cbvral 2692 . . 3  |-  ( A. y  e.  A  (
( x  e.  A  |->  B ) `  y
) R ( ( x  e.  A  |->  C ) `  y )  <->  A. x  e.  A  ( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
) )
33 simpr 109 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
343fvmpt2 5577 . . . . . 6  |-  ( ( x  e.  A  /\  B  e.  W )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3533, 1, 34syl2anc 409 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
3611fvmpt2 5577 . . . . . 6  |-  ( ( x  e.  A  /\  C  e.  X )  ->  ( ( x  e.  A  |->  C ) `  x )  =  C )
3733, 9, 36syl2anc 409 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  C ) `  x
)  =  C )
3835, 37breq12d 4000 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
)  <->  B R C ) )
3938ralbidva 2466 . . 3  |-  ( ph  ->  ( A. x  e.  A  ( ( x  e.  A  |->  B ) `
 x ) R ( ( x  e.  A  |->  C ) `  x )  <->  A. x  e.  A  B R C ) )
4032, 39syl5bb 191 . 2  |-  ( ph  ->  ( A. y  e.  A  ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y )  <->  A. x  e.  A  B R C ) )
4123, 40bitrd 187 1  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  A  B R C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   A.wral 2448   class class class wbr 3987    |-> cmpt 4048    Fn wfn 5191   ` cfv 5196    oRcofr 6057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ofr 6059
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator