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Theorem f1otrgds 24377
Description: Convenient lemma for f1otrg 24379 (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 2875 . 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 6277 . . . . 5  |-  ( e  =  X  ->  (
e E f )  =  ( X E f ) )
6 fveq2 5848 . . . . . 6  |-  ( e  =  X  ->  ( F `  e )  =  ( F `  X ) )
76oveq1d 6285 . . . . 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 6278 . . . . 5  |-  ( f  =  Y  ->  ( X E f )  =  ( X E Y ) )
10 fveq2 5848 . . . . . 6  |-  ( f  =  Y  ->  ( F `  f )  =  ( F `  Y ) )
1110oveq2d 6286 . . . . 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 3216 . . 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 659 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   Basecbs 14719   distcds 14796  Itvcitv 24033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  f1otrg  24379
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