Theorem isoselem 5867

Description: Lemma for isose 5868. (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 5042	. . . . . . . . 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 2585	. . . . . . 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 5005	. . . . . . . . . . 11 |- ( x = ( A i^i ( `' R " { z } ) ) -> ( H " x ) = ( H " ( A i^i ( `' R " { z } ) ) ) )
7	6	eleq1d 2265	. . . . . . . . . 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 2824	. . . . . . . 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 5865	. . . . . . . 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 2265	. . . . . 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 2577	. . 3 |- ( ph -> ( R Se A -> A. z e. A ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
20		isof1o 5854	. . . . 5 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
21		f1ofn 5505	. . . . 5 |- ( H : A -1-1-onto-> B -> H Fn A )
22		sneq 3633	. . . . . . . . 9 |- ( y = ( H ` z ) -> { y } = { ( H ` z ) } )
23	22	imaeq2d 5009	. . . . . . . 8 |- ( y = ( H ` z ) -> ( `' S " { y } ) = ( `' S " { ( H ` z ) } ) )
24	23	ineq2d 3364	. . . . . . 7 |- ( y = ( H ` z ) -> ( B i^i ( `' S " { y } ) ) = ( B i^i ( `' S " { ( H ` z ) } ) ) )
25	24	eleq1d 2265	. . . . . 6 |- ( y = ( H ` z ) -> ( ( B i^i ( `' S " { y } ) ) e. _V <-> ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
26	25	ralrn 5700	. . . . 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 5511	. . . . . 6 |- ( H : A -1-1-onto-> B -> H : A -onto-> B )
29		forn 5483	. . . . . 6 |- ( H : A -onto-> B -> ran H = B )
30	13, 20, 28, 29	4syl 18	. . . . 5 |- ( ph -> ran H = B )
31	30	raleqdv 2699	. . . 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 5042	. 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 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   i^i cin 3156   {csn 3622   Se wse 4364   `'ccnv 4662   ran crn 4664   "cima 4666   Fn wfn 5253   -onto->wfo 5256   -1-1-onto->wf1o 5257   ` cfv 5258   Isom wiso 5259

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-se 4368  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267

This theorem is referenced by:  isose  5868