Theorem isoselem 5999

Description: Lemma for isose 6000. (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 5140	. . . . . . . . 9 |- ( R Se A <-> A. z e. A ( A i^i ( `' R " { z } ) ) e. _V )
2	1	biimpi 120	. . . . . . . 8 |- ( R Se A -> A. z e. A ( A i^i ( `' R " { z } ) ) e. _V )
3	2	r19.21bi 2632	. . . . . . 7 |- ( ( R Se A /\ z e. A ) -> ( A i^i ( `' R " { z } ) ) e. _V )
4	3	expcom 116	. . . . . 6 |- ( z e. A -> ( R Se A -> ( A i^i ( `' R " { z } ) ) e. _V ) )
5	4	adantl 277	. . . . 5 |- ( ( ph /\ z e. A ) -> ( R Se A -> ( A i^i ( `' R " { z } ) ) e. _V ) )
6		imaeq2 5102	. . . . . . . . . . 11 |- ( x = ( A i^i ( `' R " { z } ) ) -> ( H " x ) = ( H " ( A i^i ( `' R " { z } ) ) ) )
7	6	eleq1d 2303	. . . . . . . . . 10 |- ( x = ( A i^i ( `' R " { z } ) ) -> ( ( H " x ) e. _V <-> ( H " ( A i^i ( `' R " { z } ) ) ) e. _V ) )
8	7	imbi2d 230	. . . . . . . . 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 2877	. . . . . . . 8 |- ( ( A i^i ( `' R " { z } ) ) e. _V -> ( ph -> ( H " ( A i^i ( `' R " { z } ) ) ) e. _V ) )
11	10	com12 30	. . . . . . 7 |- ( ph -> ( ( A i^i ( `' R " { z } ) ) e. _V -> ( H " ( A i^i ( `' R " { z } ) ) ) e. _V ) )
12	11	adantr 276	. . . . . 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 5997	. . . . . . . 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 283	. . . . . . 7 |- ( ( ph /\ z e. A ) -> ( H " ( A i^i ( `' R " { z } ) ) ) = ( B i^i ( `' S " { ( H ` z ) } ) ) )
16	15	eleq1d 2303	. . . . . 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 149	. . . . 5 |- ( ( ph /\ z e. A ) -> ( ( A i^i ( `' R " { z } ) ) e. _V -> ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
18	5, 17	syld 45	. . . 4 |- ( ( ph /\ z e. A ) -> ( R Se A -> ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
19	18	ralrimdva 2624	. . 3 |- ( ph -> ( R Se A -> A. z e. A ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
20		isof1o 5986	. . . . 5 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
21		f1ofn 5620	. . . . 5 |- ( H : A -1-1-onto-> B -> H Fn A )
22		sneq 3705	. . . . . . . . 9 |- ( y = ( H ` z ) -> { y } = { ( H ` z ) } )
23	22	imaeq2d 5106	. . . . . . . 8 |- ( y = ( H ` z ) -> ( `' S " { y } ) = ( `' S " { ( H ` z ) } ) )
24	23	ineq2d 3426	. . . . . . 7 |- ( y = ( H ` z ) -> ( B i^i ( `' S " { y } ) ) = ( B i^i ( `' S " { ( H ` z ) } ) ) )
25	24	eleq1d 2303	. . . . . 6 |- ( y = ( H ` z ) -> ( ( B i^i ( `' S " { y } ) ) e. _V <-> ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
26	25	ralrn 5820	. . . . 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 18	. . . 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 5626	. . . . . 6 |- ( H : A -1-1-onto-> B -> H : A -onto-> B )
29		forn 5598	. . . . . 6 |- ( H : A -onto-> B -> ran H = B )
30	13, 20, 28, 29	4syl 18	. . . . 5 |- ( ph -> ran H = B )
31	30	raleqdv 2749	. . . 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 190	. . 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 149	. 2 |- ( ph -> ( R Se A -> A. y e. B ( B i^i ( `' S " { y } ) ) e. _V ) )
34		dfse2 5140	. 2 |- ( S Se B <-> A. y e. B ( B i^i ( `' S " { y } ) ) e. _V )
35	33, 34	imbitrrdi 162	1 |- ( ph -> ( R Se A -> S Se B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   i^i cin 3213   {csn 3694   Se wse 4455   `'ccnv 4753   ran crn 4755   "cima 4757   Fn wfn 5352   -onto->wfo 5355   -1-1-onto->wf1o 5356   ` cfv 5357   Isom wiso 5358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-se 4459  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366

This theorem is referenced by:  isose  6000