Theorem isoselem 5799

Description: Lemma for isose 5800. (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 4984	. . . . . . . . 9 |- ( R Se A <-> A. z e. A ( A i^i ( `' R " { z } ) ) e. _V )
2	1	biimpi 119	. . . . . . . 8 |- ( R Se A -> A. z e. A ( A i^i ( `' R " { z } ) ) e. _V )
3	2	r19.21bi 2558	. . . . . . 7 |- ( ( R Se A /\ z e. A ) -> ( A i^i ( `' R " { z } ) ) e. _V )
4	3	expcom 115	. . . . . 6 |- ( z e. A -> ( R Se A -> ( A i^i ( `' R " { z } ) ) e. _V ) )
5	4	adantl 275	. . . . 5 |- ( ( ph /\ z e. A ) -> ( R Se A -> ( A i^i ( `' R " { z } ) ) e. _V ) )
6		imaeq2 4949	. . . . . . . . . . 11 |- ( x = ( A i^i ( `' R " { z } ) ) -> ( H " x ) = ( H " ( A i^i ( `' R " { z } ) ) ) )
7	6	eleq1d 2239	. . . . . . . . . 10 |- ( x = ( A i^i ( `' R " { z } ) ) -> ( ( H " x ) e. _V <-> ( H " ( A i^i ( `' R " { z } ) ) ) e. _V ) )
8	7	imbi2d 229	. . . . . . . . 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 2790	. . . . . . . 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 274	. . . . . 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 5797	. . . . . . . 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 281	. . . . . . 7 |- ( ( ph /\ z e. A ) -> ( H " ( A i^i ( `' R " { z } ) ) ) = ( B i^i ( `' S " { ( H ` z ) } ) ) )
16	15	eleq1d 2239	. . . . . 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 148	. . . . 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 2550	. . 3 |- ( ph -> ( R Se A -> A. z e. A ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
20		isof1o 5786	. . . . 5 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
21		f1ofn 5443	. . . . 5 |- ( H : A -1-1-onto-> B -> H Fn A )
22		sneq 3594	. . . . . . . . 9 |- ( y = ( H ` z ) -> { y } = { ( H ` z ) } )
23	22	imaeq2d 4953	. . . . . . . 8 |- ( y = ( H ` z ) -> ( `' S " { y } ) = ( `' S " { ( H ` z ) } ) )
24	23	ineq2d 3328	. . . . . . 7 |- ( y = ( H ` z ) -> ( B i^i ( `' S " { y } ) ) = ( B i^i ( `' S " { ( H ` z ) } ) ) )
25	24	eleq1d 2239	. . . . . 6 |- ( y = ( H ` z ) -> ( ( B i^i ( `' S " { y } ) ) e. _V <-> ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
26	25	ralrn 5634	. . . . 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 5449	. . . . . 6 |- ( H : A -1-1-onto-> B -> H : A -onto-> B )
29		forn 5423	. . . . . 6 |- ( H : A -onto-> B -> ran H = B )
30	13, 20, 28, 29	4syl 18	. . . . 5 |- ( ph -> ran H = B )
31	30	raleqdv 2671	. . . 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 189	. . 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 148	. 2 |- ( ph -> ( R Se A -> A. y e. B ( B i^i ( `' S " { y } ) ) e. _V ) )
34		dfse2 4984	. 2 |- ( S Se B <-> A. y e. B ( B i^i ( `' S " { y } ) ) e. _V )
35	33, 34	syl6ibr 161	1 |- ( ph -> ( R Se A -> S Se B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   i^i cin 3120   {csn 3583   Se wse 4314   `'ccnv 4610   ran crn 4612   "cima 4614   Fn wfn 5193   -onto->wfo 5196   -1-1-onto->wf1o 5197   ` cfv 5198   Isom wiso 5199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-se 4318  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207

This theorem is referenced by:  isose  5800