Theorem oismo 8998

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 5183 (the second statement is trivial under ax-rep 5183). (Contributed by Mario Carneiro, 26-Jun-2015.)

Hypothesis

Ref	Expression
oismo.1	⊢ 𝐹 = OrdIso( E , 𝐴)

Assertion

Ref	Expression
oismo	⊢ (𝐴 ⊆ 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 7491	. . . . . 6 ⊢ E We On
2		wess 5537	. . . . . 6 ⊢ (𝐴 ⊆ On → ( E We On → E We 𝐴))
3	1, 2	mpi 20	. . . . 5 ⊢ (𝐴 ⊆ On → E We 𝐴)
4		epse 5533	. . . . 5 ⊢ E Se 𝐴
5		oismo.1	. . . . . 6 ⊢ 𝐹 = OrdIso( E , 𝐴)
6	5	oiiso2 8989	. . . . 5 ⊢ (( E We 𝐴 ∧ E Se 𝐴) → 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
7	3, 4, 6	sylancl 588	. . . 4 ⊢ (𝐴 ⊆ On → 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
8	5	oicl 8987	. . . . 5 ⊢ Ord dom 𝐹
9	5	oif 8988	. . . . . . 7 ⊢ 𝐹:dom 𝐹⟶𝐴
10		frn 6515	. . . . . . 7 ⊢ (𝐹:dom 𝐹⟶𝐴 → ran 𝐹 ⊆ 𝐴)
11	9, 10	ax-mp 5	. . . . . 6 ⊢ ran 𝐹 ⊆ 𝐴
12		id 22	. . . . . 6 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ On)
13	11, 12	sstrid 3978	. . . . 5 ⊢ (𝐴 ⊆ On → ran 𝐹 ⊆ On)
14		smoiso2 8000	. . . . 5 ⊢ ((Ord dom 𝐹 ∧ ran 𝐹 ⊆ On) → ((𝐹:dom 𝐹–onto→ran 𝐹 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (dom 𝐹, ran 𝐹)))
15	8, 13, 14	sylancr 589	. . . 4 ⊢ (𝐴 ⊆ On → ((𝐹:dom 𝐹–onto→ran 𝐹 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (dom 𝐹, ran 𝐹)))
16	7, 15	mpbird 259	. . 3 ⊢ (𝐴 ⊆ On → (𝐹:dom 𝐹–onto→ran 𝐹 ∧ Smo 𝐹))
17	16	simprd 498	. 2 ⊢ (𝐴 ⊆ On → Smo 𝐹)
18	11	a1i 11	. . 3 ⊢ (𝐴 ⊆ On → ran 𝐹 ⊆ 𝐴)
19		simprl 769	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥 ∈ 𝐴)
20	3	adantr 483	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → E We 𝐴)
21	4	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → E Se 𝐴)
22		ffn 6509	. . . . . . . . . . 11 ⊢ (𝐹:dom 𝐹⟶𝐴 → 𝐹 Fn dom 𝐹)
23	9, 22	mp1i 13	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Fn dom 𝐹)
24		simplrr 776	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
25	3	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → E We 𝐴)
26	4	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → E Se 𝐴)
27		simplrl 775	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥 ∈ 𝐴)
28		simpr 487	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
29	5	oiiniseg 8991	. . . . . . . . . . . . . . 15 ⊢ ((( E We 𝐴 ∧ E Se 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝐹)) → ((𝐹‘𝑦) E 𝑥 ∨ 𝑥 ∈ ran 𝐹))
30	25, 26, 27, 28, 29	syl22anc 836	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑦) E 𝑥 ∨ 𝑥 ∈ ran 𝐹))
31	30	ord 860	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (¬ (𝐹‘𝑦) E 𝑥 → 𝑥 ∈ ran 𝐹))
32	24, 31	mt3d 150	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) E 𝑥)
33		epel 5464	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑦) E 𝑥 ↔ (𝐹‘𝑦) ∈ 𝑥)
34	32, 33	sylib 220	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ 𝑥)
35	34	ralrimiva 3182	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → ∀𝑦 ∈ dom 𝐹(𝐹‘𝑦) ∈ 𝑥)
36		ffnfv 6877	. . . . . . . . . 10 ⊢ (𝐹:dom 𝐹⟶𝑥 ↔ (𝐹 Fn dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹(𝐹‘𝑦) ∈ 𝑥))
37	23, 35, 36	sylanbrc 585	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹:dom 𝐹⟶𝑥)
38	9, 22	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝐹 Fn dom 𝐹)
39	17	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → Smo 𝐹)
40		smogt 7998	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn dom 𝐹 ∧ Smo 𝐹 ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹‘𝑦))
41	38, 39, 28, 40	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹‘𝑦))
42		ordelon 6210	. . . . . . . . . . . . . . 15 ⊢ ((Ord dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ On)
43	8, 28, 42	sylancr 589	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ On)
44		simpll 765	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝐴 ⊆ On)
45	44, 27	sseldd 3968	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥 ∈ On)
46		ontr2 6233	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ⊆ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑥))
47	43, 45, 46	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝑦 ⊆ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑥))
48	41, 34, 47	mp2and 697	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ 𝑥)
49	48	ex 415	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → (𝑦 ∈ dom 𝐹 → 𝑦 ∈ 𝑥))
50	49	ssrdv 3973	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹 ⊆ 𝑥)
51	19, 50	ssexd 5221	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹 ∈ V)
52		fex2 7632	. . . . . . . . 9 ⊢ ((𝐹:dom 𝐹⟶𝑥 ∧ dom 𝐹 ∈ V ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ V)
53	37, 51, 19, 52	syl3anc 1367	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 ∈ V)
54	5	ordtype2 8992	. . . . . . . 8 ⊢ (( E We 𝐴 ∧ E Se 𝐴 ∧ 𝐹 ∈ V) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
55	20, 21, 53, 54	syl3anc 1367	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
56		isof1o 7070	. . . . . . 7 ⊢ (𝐹 Isom E , E (dom 𝐹, 𝐴) → 𝐹:dom 𝐹–1-1-onto→𝐴)
57		f1ofo 6617	. . . . . . 7 ⊢ (𝐹:dom 𝐹–1-1-onto→𝐴 → 𝐹:dom 𝐹–onto→𝐴)
58		forn 6588	. . . . . . 7 ⊢ (𝐹:dom 𝐹–onto→𝐴 → ran 𝐹 = 𝐴)
59	55, 56, 57, 58	4syl 19	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → ran 𝐹 = 𝐴)
60	19, 59	eleqtrrd 2916	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥 ∈ ran 𝐹)
61	60	expr 459	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ ran 𝐹 → 𝑥 ∈ ran 𝐹))
62	61	pm2.18d 127	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ran 𝐹)
63	18, 62	eqelssd 3988	. 2 ⊢ (𝐴 ⊆ On → ran 𝐹 = 𝐴)
64	17, 63	jca 514	1 ⊢ (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3495   ⊆ wss 3936   class class class wbr 5059   E cep 5459   Se wse 5507   We wwe 5508  dom cdm 5550  ran crn 5551  Ord word 6185  Oncon0 6186   Fn wfn 6345  ⟶wf 6346  –onto→wfo 6348  –1-1-onto→wf1o 6349  ‘cfv 6350   Isom wiso 6351  Smo wsmo 7976  OrdIsocoi 8967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-wrecs 7941  df-smo 7977  df-recs 8002  df-oi 8968

This theorem is referenced by:  oiid  8999  hsmexlem1  9842  hsmexlem2  9843