Theorem isoselem 6591

Description: Lemma for isose 6593. (Contributed by Mario Carneiro, 23-Jun-2015.)

Hypotheses

Ref	Expression
isofrlem.1	|- ( ph -> H Isom R , S ( A , B ) )
isofrlem.2	|- ( ph -> ( H " x ) e. _V )

Assertion

Ref	Expression
isoselem	|- ( ph -> ( R Se A -> S Se B ) )

Distinct variable groups:   x, A   x, B   x, H   ph, x   x, R   x, S

Proof of Theorem isoselem

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfse2 5499	. . . . . . . . 9 |- ( R Se A <-> A. z e. A ( A i^i ( `' R " { z } ) ) e. _V )
2	1	biimpi 206	. . . . . . . 8 |- ( R Se A -> A. z e. A ( A i^i ( `' R " { z } ) ) e. _V )
3	2	r19.21bi 2932	. . . . . . 7 |- ( ( R Se A /\ z e. A ) -> ( A i^i ( `' R " { z } ) ) e. _V )
4	3	expcom 451	. . . . . 6 |- ( z e. A -> ( R Se A -> ( A i^i ( `' R " { z } ) ) e. _V ) )
5	4	adantl 482	. . . . 5 |- ( ( ph /\ z e. A ) -> ( R Se A -> ( A i^i ( `' R " { z } ) ) e. _V ) )
6		imaeq2 5462	. . . . . . . . . . 11 |- ( x = ( A i^i ( `' R " { z } ) ) -> ( H " x ) = ( H " ( A i^i ( `' R " { z } ) ) ) )
7	6	eleq1d 2686	. . . . . . . . . 10 |- ( x = ( A i^i ( `' R " { z } ) ) -> ( ( H " x ) e. _V <-> ( H " ( A i^i ( `' R " { z } ) ) ) e. _V ) )
8	7	imbi2d 330	. . . . . . . . 9 |- ( x = ( A i^i ( `' R " { z } ) ) -> ( ( ph -> ( H " x ) e. _V ) <-> ( ph -> ( H " ( A i^i ( `' R " { z } ) ) ) e. _V ) ) )
9		isofrlem.2	. . . . . . . . 9 |- ( ph -> ( H " x ) e. _V )
10	8, 9	vtoclg 3266	. . . . . . . 8 |- ( ( A i^i ( `' R " { z } ) ) e. _V -> ( ph -> ( H " ( A i^i ( `' R " { z } ) ) ) e. _V ) )
11	10	com12 32	. . . . . . 7 |- ( ph -> ( ( A i^i ( `' R " { z } ) ) e. _V -> ( H " ( A i^i ( `' R " { z } ) ) ) e. _V ) )
12	11	adantr 481	. . . . . 6 |- ( ( ph /\ z e. A ) -> ( ( A i^i ( `' R " { z } ) ) e. _V -> ( H " ( A i^i ( `' R " { z } ) ) ) e. _V ) )
13		isofrlem.1	. . . . . . . 8 |- ( ph -> H Isom R , S ( A , B ) )
14		isoini 6588	. . . . . . . 8 |- ( ( H Isom R , S ( A , B ) /\ z e. A ) -> ( H " ( A i^i ( `' R " { z } ) ) ) = ( B i^i ( `' S " { ( H ` z ) } ) ) )
15	13, 14	sylan 488	. . . . . . 7 |- ( ( ph /\ z e. A ) -> ( H " ( A i^i ( `' R " { z } ) ) ) = ( B i^i ( `' S " { ( H ` z ) } ) ) )
16	15	eleq1d 2686	. . . . . 6 |- ( ( ph /\ z e. A ) -> ( ( H " ( A i^i ( `' R " { z } ) ) ) e. _V <-> ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
17	12, 16	sylibd 229	. . . . 5 |- ( ( ph /\ z e. A ) -> ( ( A i^i ( `' R " { z } ) ) e. _V -> ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
18	5, 17	syld 47	. . . 4 |- ( ( ph /\ z e. A ) -> ( R Se A -> ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
19	18	ralrimdva 2969	. . 3 |- ( ph -> ( R Se A -> A. z e. A ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
20		isof1o 6573	. . . . 5 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
21		f1ofn 6138	. . . . 5 |- ( H : A -1-1-onto-> B -> H Fn A )
22		sneq 4187	. . . . . . . . 9 |- ( y = ( H ` z ) -> { y } = { ( H ` z ) } )
23	22	imaeq2d 5466	. . . . . . . 8 |- ( y = ( H ` z ) -> ( `' S " { y } ) = ( `' S " { ( H ` z ) } ) )
24	23	ineq2d 3814	. . . . . . 7 |- ( y = ( H ` z ) -> ( B i^i ( `' S " { y } ) ) = ( B i^i ( `' S " { ( H ` z ) } ) ) )
25	24	eleq1d 2686	. . . . . 6 |- ( y = ( H ` z ) -> ( ( B i^i ( `' S " { y } ) ) e. _V <-> ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
26	25	ralrn 6362	. . . . 5 |- ( H Fn A -> ( A. y e. ran H ( B i^i ( `' S " { y } ) ) e. _V <-> A. z e. A ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
27	13, 20, 21, 26	4syl 19	. . . 4 |- ( ph -> ( A. y e. ran H ( B i^i ( `' S " { y } ) ) e. _V <-> A. z e. A ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
28		f1ofo 6144	. . . . . 6 |- ( H : A -1-1-onto-> B -> H : A -onto-> B )
29		forn 6118	. . . . . 6 |- ( H : A -onto-> B -> ran H = B )
30	13, 20, 28, 29	4syl 19	. . . . 5 |- ( ph -> ran H = B )
31	30	raleqdv 3144	. . . 4 |- ( ph -> ( A. y e. ran H ( B i^i ( `' S " { y } ) ) e. _V <-> A. y e. B ( B i^i ( `' S " { y } ) ) e. _V ) )
32	27, 31	bitr3d 270	. . 3 |- ( ph -> ( A. z e. A ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V <-> A. y e. B ( B i^i ( `' S " { y } ) ) e. _V ) )
33	19, 32	sylibd 229	. 2 |- ( ph -> ( R Se A -> A. y e. B ( B i^i ( `' S " { y } ) ) e. _V ) )
34		dfse2 5499	. 2 |- ( S Se B <-> A. y e. B ( B i^i ( `' S " { y } ) ) e. _V )
35	33, 34	syl6ibr 242	1 |- ( ph -> ( R Se A -> S Se B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   {csn 4177   Se wse 5071   `'ccnv 5113   ran crn 5115   "cima 5117   Fn wfn 5883   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  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-sbc 3436  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-opab 4713  df-mpt 4730  df-id 5024  df-se 5074  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897

This theorem is referenced by:  isose  6593