Theorem wemoiso 7153

Description: Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 8936. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)

Assertion

Ref	Expression
wemoiso	|- ( R We A -> E* f f Isom R , S ( A , B ) )

Distinct variable groups:   R, f   A, f   S, f   B, f

Proof of Theorem wemoiso

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . 6 |- ( ( R We A /\ ( f Isom R , S ( A , B ) /\ g Isom R , S ( A , B ) ) ) -> R We A )
2		vex 3203	. . . . . . . . 9 |- f e. _V
3		isof1o 6573	. . . . . . . . . 10 |- ( f Isom R , S ( A , B ) -> f : A -1-1-onto-> B )
4		f1of 6137	. . . . . . . . . 10 |- ( f : A -1-1-onto-> B -> f : A --> B )
5	3, 4	syl 17	. . . . . . . . 9 |- ( f Isom R , S ( A , B ) -> f : A --> B )
6		dmfex 7124	. . . . . . . . 9 |- ( ( f e. _V /\ f : A --> B ) -> A e. _V )
7	2, 5, 6	sylancr 695	. . . . . . . 8 |- ( f Isom R , S ( A , B ) -> A e. _V )
8	7	ad2antrl 764	. . . . . . 7 |- ( ( R We A /\ ( f Isom R , S ( A , B ) /\ g Isom R , S ( A , B ) ) ) -> A e. _V )
9		exse 5078	. . . . . . 7 |- ( A e. _V -> R Se A )
10	8, 9	syl 17	. . . . . 6 |- ( ( R We A /\ ( f Isom R , S ( A , B ) /\ g Isom R , S ( A , B ) ) ) -> R Se A )
11	1, 10	jca 554	. . . . 5 |- ( ( R We A /\ ( f Isom R , S ( A , B ) /\ g Isom R , S ( A , B ) ) ) -> ( R We A /\ R Se A ) )
12		weisoeq 6605	. . . . 5 |- ( ( ( R We A /\ R Se A ) /\ ( f Isom R , S ( A , B ) /\ g Isom R , S ( A , B ) ) ) -> f = g )
13	11, 12	sylancom 701	. . . 4 |- ( ( R We A /\ ( f Isom R , S ( A , B ) /\ g Isom R , S ( A , B ) ) ) -> f = g )
14	13	ex 450	. . 3 |- ( R We A -> ( ( f Isom R , S ( A , B ) /\ g Isom R , S ( A , B ) ) -> f = g ) )
15	14	alrimivv 1856	. 2 |- ( R We A -> A. f A. g ( ( f Isom R , S ( A , B ) /\ g Isom R , S ( A , B ) ) -> f = g ) )
16		isoeq1 6567	. . 3 |- ( f = g -> ( f Isom R , S ( A , B ) <-> g Isom R , S ( A , B ) ) )
17	16	mo4 2517	. 2 |- ( E* f f Isom R , S ( A , B ) <-> A. f A. g ( ( f Isom R , S ( A , B ) /\ g Isom R , S ( A , B ) ) -> f = g ) )
18	15, 17	sylibr 224	1 |- ( R We A -> E* f f Isom R , S ( A , B ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   e. wcel 1990   E*wmo 2471   _Vcvv 3200   Se wse 5071   We wwe 5072   -->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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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:  fzisoeu  39514