Theorem isoselem 5490

Description: Lemma for isose 5491. (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 4728	. . . . . . . . 9 |- ( R Se A <-> A. z e. A ( A i^i ( `' R " { z } ) ) e. _V )
2	1	biimpi 118	. . . . . . . 8 |- ( R Se A -> A. z e. A ( A i^i ( `' R " { z } ) ) e. _V )
3	2	r19.21bi 2450	. . . . . . 7 |- ( ( R Se A /\ z e. A ) -> ( A i^i ( `' R " { z } ) ) e. _V )
4	3	expcom 114	. . . . . 6 |- ( z e. A -> ( R Se A -> ( A i^i ( `' R " { z } ) ) e. _V ) )
5	4	adantl 271	. . . . 5 |- ( ( ph /\ z e. A ) -> ( R Se A -> ( A i^i ( `' R " { z } ) ) e. _V ) )
6		imaeq2 4694	. . . . . . . . . . 11 |- ( x = ( A i^i ( `' R " { z } ) ) -> ( H " x ) = ( H " ( A i^i ( `' R " { z } ) ) ) )
7	6	eleq1d 2148	. . . . . . . . . 10 |- ( x = ( A i^i ( `' R " { z } ) ) -> ( ( H " x ) e. _V <-> ( H " ( A i^i ( `' R " { z } ) ) ) e. _V ) )
8	7	imbi2d 228	. . . . . . . . 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 2659	. . . . . . . 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 270	. . . . . 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 5488	. . . . . . . 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 277	. . . . . . 7 |- ( ( ph /\ z e. A ) -> ( H " ( A i^i ( `' R " { z } ) ) ) = ( B i^i ( `' S " { ( H ` z ) } ) ) )
16	15	eleq1d 2148	. . . . . 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 147	. . . . 5 |- ( ( ph /\ z e. A ) -> ( ( A i^i ( `' R " { z } ) ) e. _V -> ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
18	5, 17	syld 44	. . . 4 |- ( ( ph /\ z e. A ) -> ( R Se A -> ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
19	18	ralrimdva 2442	. . 3 |- ( ph -> ( R Se A -> A. z e. A ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
20		isof1o 5478	. . . . 5 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
21		f1ofn 5158	. . . . 5 |- ( H : A -1-1-onto-> B -> H Fn A )
22		sneq 3417	. . . . . . . . 9 |- ( y = ( H ` z ) -> { y } = { ( H ` z ) } )
23	22	imaeq2d 4698	. . . . . . . 8 |- ( y = ( H ` z ) -> ( `' S " { y } ) = ( `' S " { ( H ` z ) } ) )
24	23	ineq2d 3174	. . . . . . 7 |- ( y = ( H ` z ) -> ( B i^i ( `' S " { y } ) ) = ( B i^i ( `' S " { ( H ` z ) } ) ) )
25	24	eleq1d 2148	. . . . . 6 |- ( y = ( H ` z ) -> ( ( B i^i ( `' S " { y } ) ) e. _V <-> ( B i^i ( `' S " { ( H ` z ) } ) ) e. _V ) )
26	25	ralrn 5337	. . . . 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 5164	. . . . . 6 |- ( H : A -1-1-onto-> B -> H : A -onto-> B )
29		forn 5140	. . . . . 6 |- ( H : A -onto-> B -> ran H = B )
30	13, 20, 28, 29	4syl 18	. . . . 5 |- ( ph -> ran H = B )
31	30	raleqdv 2556	. . . 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 188	. . 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 147	. 2 |- ( ph -> ( R Se A -> A. y e. B ( B i^i ( `' S " { y } ) ) e. _V ) )
34		dfse2 4728	. 2 |- ( S Se B <-> A. y e. B ( B i^i ( `' S " { y } ) ) e. _V )
35	33, 34	syl6ibr 160	1 |- ( ph -> ( R Se A -> S Se B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434   A.wral 2349   _Vcvv 2602   i^i cin 2973   {csn 3406   Se wse 4092   `'ccnv 4370   ran crn 4372   "cima 4374   Fn wfn 4927   -onto->wfo 4930   -1-1-onto->wf1o 4931   ` cfv 4932   Isom wiso 4933

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-se 4096  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-isom 4941

This theorem is referenced by:  isose  5491