Theorem oismo 9534

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 5231 (the second statement is trivial under ax-rep 5231). (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 7789	. . . . . 6 ⊢ E We On
2		wess 5636	. . . . . 6 ⊢ (𝐴 ⊆ On → ( E We On → E We 𝐴))
3	1, 2	mpi 21	. . . . 5 ⊢ (𝐴 ⊆ On → E We 𝐴)
4		epse 5632	. . . . 5 ⊢ E Se 𝐴
5		oismo.1	. . . . . 6 ⊢ 𝐹 = OrdIso( E , 𝐴)
6	5	oiiso2 9525	. . . . 5 ⊢ (( E We 𝐴 ∧ E Se 𝐴) → 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
7	3, 4, 6	sylancl 598	. . . 4 ⊢ (𝐴 ⊆ On → 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
8	5	oicl 9523	. . . . 5 ⊢ Ord dom 𝐹
9	5	oif 9524	. . . . . . 7 ⊢ 𝐹:dom 𝐹⟶𝐴
10		frn 6717	. . . . . . 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 3919	. . . . 5 ⊢ (𝐴 ⊆ On → ran 𝐹 ⊆ On)
14		smoiso2 8377	. . . . 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 599	. . . 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 501	. 2 ⊢ (𝐴 ⊆ On → Smo 𝐹)
18	11	a1i 11	. . 3 ⊢ (𝐴 ⊆ On → ran 𝐹 ⊆ 𝐴)
19		simprl 783	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥 ∈ 𝐴)
20	3	adantr 486	. . . . . . . 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 6709	. . . . . . . . . . 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 790	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
25	3	ad2antrr 739	. . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥 ∈ 𝐴)
28		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
29	5	oiiniseg 9527	. . . . . . . . . . . . . . 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 852	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑦) E 𝑥 ∨ 𝑥 ∈ ran 𝐹))
31	30	ord 878	. . . . . . . . . . . . 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 5553	. . . . . . . . . . . 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 3154	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → ∀𝑦 ∈ dom 𝐹(𝐹‘𝑦) ∈ 𝑥)
36		ffnfv 7119	. . . . . . . . . 10 ⊢ (𝐹:dom 𝐹⟶𝑥 ↔ (𝐹 Fn dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹(𝐹‘𝑦) ∈ 𝑥))
37	23, 35, 36	sylanbrc 595	. . . . . . . . 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 739	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → Smo 𝐹)
40		smogt 8375	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn dom 𝐹 ∧ Smo 𝐹 ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹‘𝑦))
41	38, 39, 28, 40	syl3anc 1398	. . . . . . . . . . . . 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 599	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ On)
44		simpll 779	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝐴 ⊆ On)
45	44, 27	sseldd 3909	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥 ∈ On)
46		ontr2 6411	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ⊆ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑥))
47	43, 45, 46	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝑦 ⊆ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑥))
48	41, 34, 47	mp2and 712	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ 𝑥)
49	48	ex 418	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → (𝑦 ∈ dom 𝐹 → 𝑦 ∈ 𝑥))
50	49	ssrdv 3914	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹 ⊆ 𝑥)
51	19, 50	ssexd 5285	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹 ∈ V)
52		fex2 7948	. . . . . . . . 9 ⊢ ((𝐹:dom 𝐹⟶𝑥 ∧ dom 𝐹 ∈ V ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ V)
53	37, 51, 19, 52	syl3anc 1398	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 ∈ V)
54	5	ordtype2 9528	. . . . . . . 8 ⊢ (( E We 𝐴 ∧ E Se 𝐴 ∧ 𝐹 ∈ V) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
55	20, 21, 53, 54	syl3anc 1398	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
56		isof1o 7331	. . . . . . 7 ⊢ (𝐹 Isom E , E (dom 𝐹, 𝐴) → 𝐹:dom 𝐹–1-1-onto→𝐴)
57		f1ofo 6832	. . . . . . 7 ⊢ (𝐹:dom 𝐹–1-1-onto→𝐴 → 𝐹:dom 𝐹–onto→𝐴)
58		forn 6799	. . . . . . 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 2863	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥 ∈ ran 𝐹)
61	60	expr 462	. . . 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 3929	. 2 ⊢ (𝐴 ⊆ On → ran 𝐹 = 𝐴)
64	17, 63	jca 521	1 ⊢ (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   class class class wbr 5102   E cep 5549   Se wse 5601   We wwe 5602  dom cdm 5650  ran crn 5651  Ord word 6361  Oncon0 6362   Fn wfn 6533  ⟶wf 6534  –onto→wfo 6536  –1-1-onto→wf1o 6537  ‘cfv 6538   Isom wiso 6539  Smo wsmo 8353  OrdIsocoi 9503

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-smo 8354  df-recs 8379  df-oi 9504

This theorem is used by:  oiid  9535  hsmexlem1  10504  hsmexlem2  10505