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Theorem f1otrgds 23287
Description: Convenient lemma for f1otrg 23289 (Contributed by Thierry Arnoux, 19-Mar-2019.)
Hypotheses
Ref Expression
f1otrkg.p  |-  P  =  ( Base `  G
)
f1otrkg.d  |-  D  =  ( dist `  G
)
f1otrkg.i  |-  I  =  (Itv `  G )
f1otrkg.b  |-  B  =  ( Base `  H
)
f1otrkg.e  |-  E  =  ( dist `  H
)
f1otrkg.j  |-  J  =  (Itv `  H )
f1otrkg.f  |-  ( ph  ->  F : B -1-1-onto-> P )
f1otrkg.1  |-  ( (
ph  /\  ( e  e.  B  /\  f  e.  B ) )  -> 
( e E f )  =  ( ( F `  e ) D ( F `  f ) ) )
f1otrkg.2  |-  ( (
ph  /\  ( e  e.  B  /\  f  e.  B  /\  g  e.  B ) )  -> 
( g  e.  ( e J f )  <-> 
( F `  g
)  e.  ( ( F `  e ) I ( F `  f ) ) ) )
f1otrgitv.x  |-  ( ph  ->  X  e.  B )
f1otrgitv.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
f1otrgds  |-  ( ph  ->  ( X E Y )  =  ( ( F `  X ) D ( F `  Y ) ) )
Distinct variable groups:    e, f,
g, B    D, e,
f    e, E, f    e, F, f, g    e, I, f, g    e, J, f, g    e, X, f, g    ph, e,
f, g    f, Y, g
Allowed substitution hints:    D( g)    P( e, f, g)    E( g)    G( e, f, g)    H( e, f, g)    Y( e)

Proof of Theorem f1otrgds
StepHypRef Expression
1 f1otrkg.1 . . 3  |-  ( (
ph  /\  ( e  e.  B  /\  f  e.  B ) )  -> 
( e E f )  =  ( ( F `  e ) D ( F `  f ) ) )
21ralrimivva 2914 . 2  |-  ( ph  ->  A. e  e.  B  A. f  e.  B  ( e E f )  =  ( ( F `  e ) D ( F `  f ) ) )
3 f1otrgitv.x . . 3  |-  ( ph  ->  X  e.  B )
4 f1otrgitv.y . . 3  |-  ( ph  ->  Y  e.  B )
5 oveq1 6210 . . . . 5  |-  ( e  =  X  ->  (
e E f )  =  ( X E f ) )
6 fveq2 5802 . . . . . 6  |-  ( e  =  X  ->  ( F `  e )  =  ( F `  X ) )
76oveq1d 6218 . . . . 5  |-  ( e  =  X  ->  (
( F `  e
) D ( F `
 f ) )  =  ( ( F `
 X ) D ( F `  f
) ) )
85, 7eqeq12d 2476 . . . 4  |-  ( e  =  X  ->  (
( e E f )  =  ( ( F `  e ) D ( F `  f ) )  <->  ( X E f )  =  ( ( F `  X ) D ( F `  f ) ) ) )
9 oveq2 6211 . . . . 5  |-  ( f  =  Y  ->  ( X E f )  =  ( X E Y ) )
10 fveq2 5802 . . . . . 6  |-  ( f  =  Y  ->  ( F `  f )  =  ( F `  Y ) )
1110oveq2d 6219 . . . . 5  |-  ( f  =  Y  ->  (
( F `  X
) D ( F `
 f ) )  =  ( ( F `
 X ) D ( F `  Y
) ) )
129, 11eqeq12d 2476 . . . 4  |-  ( f  =  Y  ->  (
( X E f )  =  ( ( F `  X ) D ( F `  f ) )  <->  ( X E Y )  =  ( ( F `  X
) D ( F `
 Y ) ) ) )
138, 12rspc2v 3186 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. e  e.  B  A. f  e.  B  ( e E f )  =  ( ( F `  e
) D ( F `
 f ) )  ->  ( X E Y )  =  ( ( F `  X
) D ( F `
 Y ) ) ) )
143, 4, 13syl2anc 661 . 2  |-  ( ph  ->  ( A. e  e.  B  A. f  e.  B  ( e E f )  =  ( ( F `  e
) D ( F `
 f ) )  ->  ( X E Y )  =  ( ( F `  X
) D ( F `
 Y ) ) ) )
152, 14mpd 15 1  |-  ( ph  ->  ( X E Y )  =  ( ( F `  X ) D ( F `  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   -1-1-onto->wf1o 5528   ` cfv 5529  (class class class)co 6203   Basecbs 14295   distcds 14369  Itvcitv 23032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-ov 6206
This theorem is referenced by:  f1otrg  23289
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