Theorem oismo 8445

Description: When A is a subclass of On, F is a strictly monotone ordinal functions, and it is also complete (it is an isomorphism onto all of A). The proof avoids ax-rep 4771 (the second statement is trivial under ax-rep 4771). (Contributed by Mario Carneiro, 26-Jun-2015.)

Hypothesis

Ref	Expression
oismo.1	|- F = OrdIso ( _E , A )

Assertion

Ref	Expression
oismo	|- ( A C_ On -> ( Smo F /\ ran F = A ) )

Proof of Theorem oismo

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		epweon 6983	. . . . . 6 |- _E We On
2		wess 5101	. . . . . 6 |- ( A C_ On -> ( _E We On -> _E We A ) )
3	1, 2	mpi 20	. . . . 5 |- ( A C_ On -> _E We A )
4		epse 5097	. . . . 5 |- _E Se A
5		oismo.1	. . . . . 6 |- F = OrdIso ( _E , A )
6	5	oiiso2 8436	. . . . 5 |- ( ( _E We A /\ _E Se A ) -> F Isom _E , _E ( dom F , ran F ) )
7	3, 4, 6	sylancl 694	. . . 4 |- ( A C_ On -> F Isom _E , _E ( dom F , ran F ) )
8	5	oicl 8434	. . . . 5 |- Ord dom F
9	5	oif 8435	. . . . . . 7 |- F : dom F --> A
10		frn 6053	. . . . . . 7 |- ( F : dom F --> A -> ran F C_ A )
11	9, 10	ax-mp 5	. . . . . 6 |- ran F C_ A
12		id 22	. . . . . 6 |- ( A C_ On -> A C_ On )
13	11, 12	syl5ss 3614	. . . . 5 |- ( A C_ On -> ran F C_ On )
14		smoiso2 7466	. . . . 5 |- ( ( Ord dom F /\ ran F C_ On ) -> ( ( F : dom F -onto-> ran F /\ Smo F ) <-> F Isom _E , _E ( dom F , ran F ) ) )
15	8, 13, 14	sylancr 695	. . . 4 |- ( A C_ On -> ( ( F : dom F -onto-> ran F /\ Smo F ) <-> F Isom _E , _E ( dom F , ran F ) ) )
16	7, 15	mpbird 247	. . 3 |- ( A C_ On -> ( F : dom F -onto-> ran F /\ Smo F ) )
17	16	simprd 479	. 2 |- ( A C_ On -> Smo F )
18	11	a1i 11	. . 3 |- ( A C_ On -> ran F C_ A )
19		simprl 794	. . . . . . . 8 |- ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) -> x e. A )
20	3	adantr 481	. . . . . . . . . 10 |- ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) -> _E We A )
21	4	a1i 11	. . . . . . . . . 10 |- ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) -> _E Se A )
22		ffn 6045	. . . . . . . . . . . . 13 |- ( F : dom F --> A -> F Fn dom F )
23	9, 22	mp1i 13	. . . . . . . . . . . 12 |- ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) -> F Fn dom F )
24		simplrr 801	. . . . . . . . . . . . . . 15 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> -. x e. ran F )
25	3	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> _E We A )
26	4	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> _E Se A )
27		simplrl 800	. . . . . . . . . . . . . . . . 17 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> x e. A )
28		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> y e. dom F )
29	5	oiiniseg 8438	. . . . . . . . . . . . . . . . 17 |- ( ( ( _E We A /\ _E Se A ) /\ ( x e. A /\ y e. dom F ) ) -> ( ( F ` y ) _E x \/ x e. ran F ) )
30	25, 26, 27, 28, 29	syl22anc 1327	. . . . . . . . . . . . . . . 16 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> ( ( F ` y ) _E x \/ x e. ran F ) )
31	30	ord 392	. . . . . . . . . . . . . . 15 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> ( -. ( F ` y ) _E x -> x e. ran F ) )
32	24, 31	mt3d 140	. . . . . . . . . . . . . 14 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> ( F ` y ) _E x )
33		vex 3203	. . . . . . . . . . . . . . 15 |- x e. _V
34	33	epelc 5031	. . . . . . . . . . . . . 14 |- ( ( F ` y ) _E x <-> ( F ` y ) e. x )
35	32, 34	sylib 208	. . . . . . . . . . . . 13 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> ( F ` y ) e. x )
36	35	ralrimiva 2966	. . . . . . . . . . . 12 |- ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) -> A. y e. dom F ( F ` y ) e. x )
37		ffnfv 6388	. . . . . . . . . . . 12 |- ( F : dom F --> x <-> ( F Fn dom F /\ A. y e. dom F ( F ` y ) e. x ) )
38	23, 36, 37	sylanbrc 698	. . . . . . . . . . 11 |- ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) -> F : dom F --> x )
39	9, 22	mp1i 13	. . . . . . . . . . . . . . . 16 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> F Fn dom F )
40	17	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> Smo F )
41		smogt 7464	. . . . . . . . . . . . . . . 16 |- ( ( F Fn dom F /\ Smo F /\ y e. dom F ) -> y C_ ( F ` y ) )
42	39, 40, 28, 41	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> y C_ ( F ` y ) )
43		ordelon 5747	. . . . . . . . . . . . . . . . 17 |- ( ( Ord dom F /\ y e. dom F ) -> y e. On )
44	8, 28, 43	sylancr 695	. . . . . . . . . . . . . . . 16 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> y e. On )
45		simpll 790	. . . . . . . . . . . . . . . . 17 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> A C_ On )
46	45, 27	sseldd 3604	. . . . . . . . . . . . . . . 16 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> x e. On )
47		ontr2 5772	. . . . . . . . . . . . . . . 16 |- ( ( y e. On /\ x e. On ) -> ( ( y C_ ( F ` y ) /\ ( F ` y ) e. x ) -> y e. x ) )
48	44, 46, 47	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> ( ( y C_ ( F ` y ) /\ ( F ` y ) e. x ) -> y e. x ) )
49	42, 35, 48	mp2and 715	. . . . . . . . . . . . . 14 |- ( ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) /\ y e. dom F ) -> y e. x )
50	49	ex 450	. . . . . . . . . . . . 13 |- ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) -> ( y e. dom F -> y e. x ) )
51	50	ssrdv 3609	. . . . . . . . . . . 12 |- ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) -> dom F C_ x )
52	19, 51	ssexd 4805	. . . . . . . . . . 11 |- ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) -> dom F e. _V )
53		fex2 7121	. . . . . . . . . . 11 |- ( ( F : dom F --> x /\ dom F e. _V /\ x e. A ) -> F e. _V )
54	38, 52, 19, 53	syl3anc 1326	. . . . . . . . . 10 |- ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) -> F e. _V )
55	5	ordtype2 8439	. . . . . . . . . 10 |- ( ( _E We A /\ _E Se A /\ F e. _V ) -> F Isom _E , _E ( dom F , A ) )
56	20, 21, 54, 55	syl3anc 1326	. . . . . . . . 9 |- ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) -> F Isom _E , _E ( dom F , A ) )
57		isof1o 6573	. . . . . . . . 9 |- ( F Isom _E , _E ( dom F , A ) -> F : dom F -1-1-onto-> A )
58		f1ofo 6144	. . . . . . . . 9 |- ( F : dom F -1-1-onto-> A -> F : dom F -onto-> A )
59		forn 6118	. . . . . . . . 9 |- ( F : dom F -onto-> A -> ran F = A )
60	56, 57, 58, 59	4syl 19	. . . . . . . 8 |- ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) -> ran F = A )
61	19, 60	eleqtrrd 2704	. . . . . . 7 |- ( ( A C_ On /\ ( x e. A /\ -. x e. ran F ) ) -> x e. ran F )
62	61	expr 643	. . . . . 6 |- ( ( A C_ On /\ x e. A ) -> ( -. x e. ran F -> x e. ran F ) )
63	62	pm2.18d 124	. . . . 5 |- ( ( A C_ On /\ x e. A ) -> x e. ran F )
64	63	ex 450	. . . 4 |- ( A C_ On -> ( x e. A -> x e. ran F ) )
65	64	ssrdv 3609	. . 3 |- ( A C_ On -> A C_ ran F )
66	18, 65	eqssd 3620	. 2 |- ( A C_ On -> ran F = A )
67	17, 66	jca 554	1 |- ( A C_ On -> ( Smo F /\ ran F = A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   _E cep 5028   Se wse 5071   We wwe 5072   dom cdm 5114   ran crn 5115   Ord word 5722   Oncon0 5723   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889   Smo wsmo 7442  OrdIsocoi 8414

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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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  df-riota 6611  df-wrecs 7407  df-smo 7443  df-recs 7468  df-oi 8415

This theorem is referenced by:  oiid  8446  hsmexlem1  9248  hsmexlem2  9249