Theorem f1otrgds 25749

Description: Convenient lemma for f1otrg 25751. (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

Step	Hyp	Ref	Expression
1		f1otrkg.1	. . 3 |- ( ( ph /\ ( e e. B /\ f e. B ) ) -> ( e E f ) = ( ( F ` e ) D ( F ` f ) ) )
2	1	ralrimivva 2971	. 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 6657	. . . . 5 |- ( e = X -> ( e E f ) = ( X E f ) )
6		fveq2 6191	. . . . . 6 |- ( e = X -> ( F ` e ) = ( F ` X ) )
7	6	oveq1d 6665	. . . . 5 |- ( e = X -> ( ( F ` e ) D ( F ` f ) ) = ( ( F ` X ) D ( F ` f ) ) )
8	5, 7	eqeq12d 2637	. . . 4 |- ( e = X -> ( ( e E f ) = ( ( F ` e ) D ( F ` f ) ) <-> ( X E f ) = ( ( F ` X ) D ( F ` f ) ) ) )
9		oveq2 6658	. . . . 5 |- ( f = Y -> ( X E f ) = ( X E Y ) )
10		fveq2 6191	. . . . . 6 |- ( f = Y -> ( F ` f ) = ( F ` Y ) )
11	10	oveq2d 6666	. . . . 5 |- ( f = Y -> ( ( F ` X ) D ( F ` f ) ) = ( ( F ` X ) D ( F ` Y ) ) )
12	9, 11	eqeq12d 2637	. . . 4 |- ( f = Y -> ( ( X E f ) = ( ( F ` X ) D ( F ` f ) ) <-> ( X E Y ) = ( ( F ` X ) D ( F ` Y ) ) ) )
13	8, 12	rspc2v 3322	. . 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 ) ) ) )
14	3, 4, 13	syl2anc 693	. 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 ) ) ) )
15	2, 14	mpd 15	1 |- ( ph -> ( X E Y ) = ( ( F ` X ) D ( F ` Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  Itvcitv 25335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  f1otrg  25751