Theorem oieu 7254

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 733	. . . . . 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 7247	. . . . . . . 8 |- ( ( R We A /\ R Se A ) -> F Isom _E , R ( dom F , A ) )
4	3	adantr 451	. . . . . . 7 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> F Isom _E , R ( dom F , A ) )
5		isocnv 5827	. . . . . . 7 |- ( F Isom _E , R ( dom F , A ) -> `' F Isom R , _E ( A , dom F ) )
6	4, 5	syl 15	. . . . . 6 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> `' F Isom R , _E ( A , dom F ) )
7		isotr 5833	. . . . . 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 642	. . . . 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 732	. . . . 5 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> Ord B )
10	2	oicl 7244	. . . . . 6 |- Ord dom F
11	10	a1i 10	. . . . 5 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> Ord dom F )
12		ordiso2 7230	. . . . 5 |- ( ( ( `' F o. G ) Isom _E , _E ( B , dom F ) /\ Ord B /\ Ord dom F ) -> B = dom F )
13	8, 9, 11, 12	syl3anc 1182	. . . 4 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> B = dom F )
14		ordwe 4405	. . . . . 6 |- ( Ord B -> _E We B )
15	14	ad2antrl 708	. . . . 5 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> _E We B )
16		epse 4376	. . . . . 6 |- _E Se B
17	16	a1i 10	. . . . 5 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> _E Se B )
18		isoeq4 5819	. . . . . . 7 |- ( B = dom F -> ( F Isom _E , R ( B , A ) <-> F Isom _E , R ( dom F , A ) ) )
19	13, 18	syl 15	. . . . . 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 223	. . . . 5 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> F Isom _E , R ( B , A ) )
21		weisoeq 5853	. . . . 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 1183	. . . 4 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> G = F )
23	13, 22	jca 518	. . 3 |- ( ( ( R We A /\ R Se A ) /\ ( Ord B /\ G Isom _E , R ( B , A ) ) ) -> ( B = dom F /\ G = F ) )
24	23	ex 423	. 2 |- ( ( R We A /\ R Se A ) -> ( ( Ord B /\ G Isom _E , R ( B , A ) ) -> ( B = dom F /\ G = F ) ) )
25	3, 10	jctil 523	. . 3 |- ( ( R We A /\ R Se A ) -> ( Ord dom F /\ F Isom _E , R ( dom F , A ) ) )
26		ordeq 4399	. . . . 5 |- ( B = dom F -> ( Ord B <-> Ord dom F ) )
27	26	adantr 451	. . . 4 |- ( ( B = dom F /\ G = F ) -> ( Ord B <-> Ord dom F ) )
28		isoeq4 5819	. . . . 5 |- ( B = dom F -> ( G Isom _E , R ( B , A ) <-> G Isom _E , R ( dom F , A ) ) )
29		isoeq1 5816	. . . . 5 |- ( G = F -> ( G Isom _E , R ( dom F , A ) <-> F Isom _E , R ( dom F , A ) ) )
30	28, 29	sylan9bb 680	. . . 4 |- ( ( B = dom F /\ G = F ) -> ( G Isom _E , R ( B , A ) <-> F Isom _E , R ( dom F , A ) ) )
31	27, 30	anbi12d 691	. . 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 213	. 2 |- ( ( R We A /\ R Se A ) -> ( ( B = dom F /\ G = F ) -> ( Ord B /\ G Isom _E , R ( B , A ) ) ) )
33	24, 32	impbid 183	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 176   /\ wa 358   = wceq 1623   _E cep 4303   Se wse 4350   We wwe 4351   Ord word 4391   `'ccnv 4688   dom cdm 4689   o. ccom 4693   Isom wiso 5256  OrdIsocoi 7224

This theorem is referenced by:  hartogslem1  7257  cantnfp1lem3  7382  oemapwe  7396  cantnffval2  7397  om2uzoi  11018

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-oi 7225