Theorem isofr2 6594

Description: A weak form of isofr 6592 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)

Assertion

Ref	Expression
isofr2	|- ( ( H Isom R , S ( A , B ) /\ B e. V ) -> ( S Fr B -> R Fr A ) )

Proof of Theorem isofr2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. 2 |- ( ( H Isom R , S ( A , B ) /\ B e. V ) -> H Isom R , S ( A , B ) )
2		imassrn 5477	. . . 4 |- ( H " x ) C_ ran H
3		isof1o 6573	. . . . 5 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
4		f1of 6137	. . . . 5 |- ( H : A -1-1-onto-> B -> H : A --> B )
5		frn 6053	. . . . 5 |- ( H : A --> B -> ran H C_ B )
6	3, 4, 5	3syl 18	. . . 4 |- ( H Isom R , S ( A , B ) -> ran H C_ B )
7	2, 6	syl5ss 3614	. . 3 |- ( H Isom R , S ( A , B ) -> ( H " x ) C_ B )
8		ssexg 4804	. . 3 |- ( ( ( H " x ) C_ B /\ B e. V ) -> ( H " x ) e. _V )
9	7, 8	sylan 488	. 2 |- ( ( H Isom R , S ( A , B ) /\ B e. V ) -> ( H " x ) e. _V )
10	1, 9	isofrlem 6590	1 |- ( ( H Isom R , S ( A , B ) /\ B e. V ) -> ( S Fr B -> R Fr A ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   C_ wss 3574   Fr wfr 5070   ran crn 5115   "cima 5117   -->wf 5884   -1-1-onto->wf1o 5887   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-ne 2795  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-id 5024  df-fr 5073  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: (None)