Theorem oismo 9532

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 5285 (the second statement is trivial under ax-rep 5285). (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 7759	. . . . . 6 ⊢ E We On
2		wess 5663	. . . . . 6 ⊢ (𝐴 ⊆ On → ( E We On → E We 𝐴))
3	1, 2	mpi 20	. . . . 5 ⊢ (𝐴 ⊆ On → E We 𝐴)
4		epse 5659	. . . . 5 ⊢ E Se 𝐴
5		oismo.1	. . . . . 6 ⊢ 𝐹 = OrdIso( E , 𝐴)
6	5	oiiso2 9523	. . . . 5 ⊢ (( E We 𝐴 ∧ E Se 𝐴) → 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
7	3, 4, 6	sylancl 587	. . . 4 ⊢ (𝐴 ⊆ On → 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
8	5	oicl 9521	. . . . 5 ⊢ Ord dom 𝐹
9	5	oif 9522	. . . . . . 7 ⊢ 𝐹:dom 𝐹⟶𝐴
10		frn 6722	. . . . . . 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 3993	. . . . 5 ⊢ (𝐴 ⊆ On → ran 𝐹 ⊆ On)
14		smoiso2 8366	. . . . 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 588	. . . 4 ⊢ (𝐴 ⊆ On → ((𝐹:dom 𝐹–onto→ran 𝐹 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (dom 𝐹, ran 𝐹)))
16	7, 15	mpbird 257	. . 3 ⊢ (𝐴 ⊆ On → (𝐹:dom 𝐹–onto→ran 𝐹 ∧ Smo 𝐹))
17	16	simprd 497	. 2 ⊢ (𝐴 ⊆ On → Smo 𝐹)
18	11	a1i 11	. . 3 ⊢ (𝐴 ⊆ On → ran 𝐹 ⊆ 𝐴)
19		simprl 770	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥 ∈ 𝐴)
20	3	adantr 482	. . . . . . . 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 6715	. . . . . . . . . . 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 777	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
25	3	ad2antrr 725	. . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥 ∈ 𝐴)
28		simpr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
29	5	oiiniseg 9525	. . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑦) E 𝑥 ∨ 𝑥 ∈ ran 𝐹))
31	30	ord 863	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (¬ (𝐹‘𝑦) E 𝑥 → 𝑥 ∈ ran 𝐹))
32	24, 31	mt3d 148	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) E 𝑥)
33		epel 5583	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑦) E 𝑥 ↔ (𝐹‘𝑦) ∈ 𝑥)
34	32, 33	sylib 217	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ 𝑥)
35	34	ralrimiva 3147	. . . . . . . . . 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 584	. . . . . . . . 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 725	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → Smo 𝐹)
40		smogt 8364	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn dom 𝐹 ∧ Smo 𝐹 ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹‘𝑦))
41	38, 39, 28, 40	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹‘𝑦))
42		ordelon 6386	. . . . . . . . . . . . . . 15 ⊢ ((Ord dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ On)
43	8, 28, 42	sylancr 588	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ On)
44		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝐴 ⊆ On)
45	44, 27	sseldd 3983	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥 ∈ On)
46		ontr2 6409	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ⊆ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑥))
47	43, 45, 46	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝑦 ⊆ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑥))
48	41, 34, 47	mp2and 698	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ 𝑥)
49	48	ex 414	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → (𝑦 ∈ dom 𝐹 → 𝑦 ∈ 𝑥))
50	49	ssrdv 3988	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹 ⊆ 𝑥)
51	19, 50	ssexd 5324	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹 ∈ V)
52		fex2 7921	. . . . . . . . 9 ⊢ ((𝐹:dom 𝐹⟶𝑥 ∧ dom 𝐹 ∈ V ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ V)
53	37, 51, 19, 52	syl3anc 1372	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 ∈ V)
54	5	ordtype2 9526	. . . . . . . 8 ⊢ (( E We 𝐴 ∧ E Se 𝐴 ∧ 𝐹 ∈ V) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
55	20, 21, 53, 54	syl3anc 1372	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
56		isof1o 7317	. . . . . . 7 ⊢ (𝐹 Isom E , E (dom 𝐹, 𝐴) → 𝐹:dom 𝐹–1-1-onto→𝐴)
57		f1ofo 6838	. . . . . . 7 ⊢ (𝐹:dom 𝐹–1-1-onto→𝐴 → 𝐹:dom 𝐹–onto→𝐴)
58		forn 6806	. . . . . . 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 2837	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥 ∈ ran 𝐹)
61	60	expr 458	. . . 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 4003	. 2 ⊢ (𝐴 ⊆ On → ran 𝐹 = 𝐴)
64	17, 63	jca 513	1 ⊢ (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ⊆ wss 3948   class class class wbr 5148   E cep 5579   Se wse 5629   We wwe 5630  dom cdm 5676  ran crn 5677  Ord word 6361  Oncon0 6362   Fn wfn 6536  ⟶wf 6537  –onto→wfo 6539  –1-1-onto→wf1o 6540  ‘cfv 6541   Isom wiso 6542  Smo wsmo 8342  OrdIsocoi 9501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-smo 8343  df-recs 8368  df-oi 9502

This theorem is referenced by:  oiid  9533  hsmexlem1  10418  hsmexlem2  10419