Theorem oismo 9502

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 5240 (the second statement is trivial under ax-rep 5240). (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 7774	. . . . . 6 ⊢ E We On
2		wess 5648	. . . . . 6 ⊢ (𝐴 ⊆ On → ( E We On → E We 𝐴))
3	1, 2	mpi 21	. . . . 5 ⊢ (𝐴 ⊆ On → E We 𝐴)
4		epse 5644	. . . . 5 ⊢ E Se 𝐴
5		oismo.1	. . . . . 6 ⊢ 𝐹 = OrdIso( E , 𝐴)
6	5	oiiso2 9493	. . . . 5 ⊢ (( E We 𝐴 ∧ E Se 𝐴) → 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
7	3, 4, 6	sylancl 597	. . . 4 ⊢ (𝐴 ⊆ On → 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
8	5	oicl 9491	. . . . 5 ⊢ Ord dom 𝐹
9	5	oif 9492	. . . . . . 7 ⊢ 𝐹:dom 𝐹⟶𝐴
10		frn 6714	. . . . . . 7 ⊢ (𝐹:dom 𝐹⟶𝐴 → ran 𝐹 ⊆ 𝐴)
11	9, 10	ax-mp 5	. . . . . 6 ⊢ ran 𝐹 ⊆ 𝐴
12		id 23	. . . . . 6 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ On)
13	11, 12	sstrid 3954	. . . . 5 ⊢ (𝐴 ⊆ On → ran 𝐹 ⊆ On)
14		smoiso2 8356	. . . . 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 598	. . . 4 ⊢ (𝐴 ⊆ On → ((𝐹:dom 𝐹–onto→ran 𝐹 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (dom 𝐹, ran 𝐹)))
16	7, 15	mpbird 260	. . 3 ⊢ (𝐴 ⊆ On → (𝐹:dom 𝐹–onto→ran 𝐹 ∧ Smo 𝐹))
17	16	simprd 500	. 2 ⊢ (𝐴 ⊆ On → Smo 𝐹)
18	11	a1i 11	. . 3 ⊢ (𝐴 ⊆ On → ran 𝐹 ⊆ 𝐴)
19		simprl 782	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥 ∈ 𝐴)
20	3	adantr 485	. . . . . . . 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 6706	. . . . . . . . . . 11 ⊢ (𝐹:dom 𝐹⟶𝐴 → 𝐹 Fn dom 𝐹)
23	9, 22	mp1i 14	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Fn dom 𝐹)
24		simplrr 789	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
25	3	ad2antrr 738	. . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥 ∈ 𝐴)
28		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
29	5	oiiniseg 9495	. . . . . . . . . . . . . . 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 851	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑦) E 𝑥 ∨ 𝑥 ∈ ran 𝐹))
31	30	ord 877	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (¬ (𝐹‘𝑦) E 𝑥 → 𝑥 ∈ ran 𝐹))
32	24, 31	mt3d 149	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) E 𝑥)
33		epel 5565	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑦) E 𝑥 ↔ (𝐹‘𝑦) ∈ 𝑥)
34	32, 33	sylib 221	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ 𝑥)
35	34	ralrimiva 3163	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → ∀𝑦 ∈ dom 𝐹(𝐹‘𝑦) ∈ 𝑥)
36		ffnfv 7115	. . . . . . . . . 10 ⊢ (𝐹:dom 𝐹⟶𝑥 ↔ (𝐹 Fn dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹(𝐹‘𝑦) ∈ 𝑥))
37	23, 35, 36	sylanbrc 594	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹:dom 𝐹⟶𝑥)
38	9, 22	mp1i 14	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝐹 Fn dom 𝐹)
39	17	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → Smo 𝐹)
40		smogt 8354	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn dom 𝐹 ∧ Smo 𝐹 ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹‘𝑦))
41	38, 39, 28, 40	syl3anc 1396	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹‘𝑦))
42		ordelon 6385	. . . . . . . . . . . . . . 15 ⊢ ((Ord dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ On)
43	8, 28, 42	sylancr 598	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ On)
44		simpll 778	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝐴 ⊆ On)
45	44, 27	sseldd 3944	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥 ∈ On)
46		ontr2 6410	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ⊆ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑥))
47	43, 45, 46	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝑦 ⊆ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑥))
48	41, 34, 47	mp2and 711	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ 𝑥)
49	48	ex 417	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → (𝑦 ∈ dom 𝐹 → 𝑦 ∈ 𝑥))
50	49	ssrdv 3949	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹 ⊆ 𝑥)
51	19, 50	ssexd 5295	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹 ∈ V)
52		fex2 7933	. . . . . . . . 9 ⊢ ((𝐹:dom 𝐹⟶𝑥 ∧ dom 𝐹 ∈ V ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ V)
53	37, 51, 19, 52	syl3anc 1396	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 ∈ V)
54	5	ordtype2 9496	. . . . . . . 8 ⊢ (( E We 𝐴 ∧ E Se 𝐴 ∧ 𝐹 ∈ V) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
55	20, 21, 53, 54	syl3anc 1396	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
56		isof1o 7322	. . . . . . 7 ⊢ (𝐹 Isom E , E (dom 𝐹, 𝐴) → 𝐹:dom 𝐹–1-1-onto→𝐴)
57		f1ofo 6829	. . . . . . 7 ⊢ (𝐹:dom 𝐹–1-1-onto→𝐴 → 𝐹:dom 𝐹–onto→𝐴)
58		forn 6796	. . . . . . 7 ⊢ (𝐹:dom 𝐹–onto→𝐴 → ran 𝐹 = 𝐴)
59	55, 56, 57, 58	4syl 20	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → ran 𝐹 = 𝐴)
60	19, 59	eleqtrrd 2872	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥 ∈ ran 𝐹)
61	60	expr 461	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ ran 𝐹 → 𝑥 ∈ ran 𝐹))
62	61	pm2.18d 128	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ran 𝐹)
63	18, 62	eqelssd 3964	. 2 ⊢ (𝐴 ⊆ On → ran 𝐹 = 𝐴)
64	17, 63	jca 520	1 ⊢ (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3461   ⊆ wss 3911   class class class wbr 5111   E cep 5561   Se wse 5613   We wwe 5614  dom cdm 5662  ran crn 5663  Ord word 6360  Oncon0 6361   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537   Isom wiso 6538  Smo wsmo 8332  OrdIsocoi 9471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-smo 8333  df-recs 8358  df-oi 9472

This theorem is referenced by:  oiid  9503  hsmexlem1  10410  hsmexlem2  10411