Theorem oieu 7464

Description: Uniqueness of the unique ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)

Hypothesis

Ref	Expression
oicl.1	|- F = OrdIso ( R , A )

Assertion

Ref	Expression
oieu	|- ( ( R We A /\ R Se A ) -> ( ( Ord B /\ G Isom _E , R ( B , A ) ) <-> ( B = dom F /\ G = F ) ) )

Proof of Theorem oieu

Step	Hyp	Ref	Expression
1		simprr 734	. . . . . 6 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> G Isom _E , R ( B , A ) )
2		oicl.1	. . . . . . . . 9 |- F = OrdIso ( R , A )
3	2	ordtype 7457	. . . . . . . 8 |- ( ( R We A /\ R Se A ) -> F Isom _E , R ( dom F , A ) )
4	3	adantr 452	. . . . . . 7 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> F Isom _E , R ( dom F , A ) )
5		isocnv 6009	. . . . . . 7 |- ( F Isom _E , R ( dom F , A ) -> `' F Isom R , _E ( A , dom F ) )
6	4, 5	syl 16	. . . . . 6 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> `' F Isom R , _E ( A , dom F ) )
7		isotr 6015	. . . . . 6 |- ( ( G Isom _E , R ( B , A ) /\ `' F Isom R , _E ( A , dom F ) ) -> ( `' F o. G ) Isom _E , _E ( B , dom F ) )
8	1, 6, 7	syl2anc 643	. . . . 5 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> ( `' F o. G ) Isom _E , _E ( B , dom F ) )
9		simprl 733	. . . . 5 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> Ord B )
10	2	oicl 7454	. . . . . 6 |- Ord dom F
11	10	a1i 11	. . . . 5 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> Ord dom F )
12		ordiso2 7440	. . . . 5 |- ( ( ( `' F o. G ) Isom _E , _E ( B , dom F ) /\ Ord B /\ Ord dom F ) -> B = dom F )
13	8, 9, 11, 12	syl3anc 1184	. . . 4 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> B = dom F )
14		ordwe 4554	. . . . . 6 |- ( Ord B -> _E We B )
15	14	ad2antrl 709	. . . . 5 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> _E We B )
16		epse 4525	. . . . . 6 |- _E Se B
17	16	a1i 11	. . . . 5 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> _E Se B )
18		isoeq4 6001	. . . . . . 7 |- ( B = dom F -> ( F Isom _E , R ( B , A ) <-> F Isom _E , R ( dom F , A ) ) )
19	13, 18	syl 16	. . . . . 6 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> ( F Isom _E , R ( B , A ) <-> F Isom _E , R ( dom F , A ) ) )
20	4, 19	mpbird 224	. . . . 5 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> F Isom _E , R ( B , A ) )
21		weisoeq 6035	. . . . 5 |- ( ( ( _E We B /\ _E Se B ) /\ ( G Isom _E , R ( B , A ) /\ F Isom _E , R ( B , A ) ) ) -> G = F )
22	15, 17, 1, 20, 21	syl22anc 1185	. . . 4 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> G = F )
23	13, 22	jca 519	. . 3 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> ( B = dom F /\ G = F ) )
24	23	ex 424	. 2 |- ( ( R We A /\ R Se A ) -> ( ( Ord B /\ G Isom _E , R ( B , A ) ) -> ( B = dom F /\ G = F ) ) )
25	3, 10	jctil 524	. . 3 |- ( ( R We A /\ R Se A ) -> ( Ord dom F /\ F Isom _E , R ( dom F , A ) ) )
26		ordeq 4548	. . . . 5 |- ( B = dom F -> ( Ord B <-> Ord dom F ) )
27	26	adantr 452	. . . 4 |- ( ( B = dom F /\ G = F ) -> ( Ord B <-> Ord dom F ) )
28		isoeq4 6001	. . . . 5 |- ( B = dom F -> ( G Isom _E , R ( B , A ) <-> G Isom _E , R ( dom F , A ) ) )
29		isoeq1 5998	. . . . 5 |- ( G = F -> ( G Isom _E , R ( dom F , A ) <-> F Isom _E , R ( dom F , A ) ) )
30	28, 29	sylan9bb 681	. . . 4 |- ( ( B = dom F /\ G = F ) -> ( G Isom _E , R ( B , A ) <-> F Isom _E , R ( dom F , A ) ) )
31	27, 30	anbi12d 692	. . 3 |- ( ( B = dom F /\ G = F ) -> ( ( Ord B /\ G Isom _E , R ( B , A ) ) <-> ( Ord dom F /\ F Isom _E , R ( dom F , A ) ) ) )
32	25, 31	syl5ibrcom 214	. 2 |- ( ( R We A /\ R Se A ) -> ( ( B = dom F /\ G = F ) -> ( Ord B /\ G Isom _E , R ( B , A ) ) ) )
33	24, 32	impbid 184	1 |- ( ( R We A /\ R Se A ) -> ( ( Ord B /\ G Isom _E , R ( B , A ) ) <-> ( B = dom F /\ G = F ) ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   _E cep 4452   Se wse 4499   We wwe 4500   Ord word 4540   `'ccnv 4836   dom cdm 4837   o. ccom 4841   Isom wiso 5414  OrdIsocoi 7434

This theorem is referenced by:  hartogslem1  7467  cantnfp1lem3  7592  oemapwe  7606  cantnffval2  7607  om2uzoi  11250

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6508  df-recs 6592  df-oi 7435