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Theorem oismo 9580
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 5279 (the second statement is trivial under ax-rep 5279). (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.
StepHypRef Expression
1 epweon 7795 . . . . . 6 E We On
2 wess 5671 . . . . . 6 (𝐴 ⊆ On → ( E We On → E We 𝐴))
31, 2mpi 20 . . . . 5 (𝐴 ⊆ On → E We 𝐴)
4 epse 5667 . . . . 5 E Se 𝐴
5 oismo.1 . . . . . 6 𝐹 = OrdIso( E , 𝐴)
65oiiso2 9571 . . . . 5 (( E We 𝐴 ∧ E Se 𝐴) → 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
73, 4, 6sylancl 586 . . . 4 (𝐴 ⊆ On → 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
85oicl 9569 . . . . 5 Ord dom 𝐹
95oif 9570 . . . . . . 7 𝐹:dom 𝐹𝐴
10 frn 6743 . . . . . . 7 (𝐹:dom 𝐹𝐴 → ran 𝐹𝐴)
119, 10ax-mp 5 . . . . . 6 ran 𝐹𝐴
12 id 22 . . . . . 6 (𝐴 ⊆ On → 𝐴 ⊆ On)
1311, 12sstrid 3995 . . . . 5 (𝐴 ⊆ On → ran 𝐹 ⊆ On)
14 smoiso2 8409 . . . . 5 ((Ord dom 𝐹 ∧ ran 𝐹 ⊆ On) → ((𝐹:dom 𝐹onto→ran 𝐹 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (dom 𝐹, ran 𝐹)))
158, 13, 14sylancr 587 . . . 4 (𝐴 ⊆ On → ((𝐹:dom 𝐹onto→ran 𝐹 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (dom 𝐹, ran 𝐹)))
167, 15mpbird 257 . . 3 (𝐴 ⊆ On → (𝐹:dom 𝐹onto→ran 𝐹 ∧ Smo 𝐹))
1716simprd 495 . 2 (𝐴 ⊆ On → Smo 𝐹)
1811a1i 11 . . 3 (𝐴 ⊆ On → ran 𝐹𝐴)
19 simprl 771 . . . . . 6 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥𝐴)
203adantr 480 . . . . . . . 8 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → E We 𝐴)
214a1i 11 . . . . . . . 8 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → E Se 𝐴)
22 ffn 6736 . . . . . . . . . . 11 (𝐹:dom 𝐹𝐴𝐹 Fn dom 𝐹)
239, 22mp1i 13 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Fn dom 𝐹)
24 simplrr 778 . . . . . . . . . . . . 13 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
253ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → E We 𝐴)
264a1i 11 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → E Se 𝐴)
27 simplrl 777 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥𝐴)
28 simpr 484 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
295oiiniseg 9573 . . . . . . . . . . . . . . 15 ((( E We 𝐴 ∧ E Se 𝐴) ∧ (𝑥𝐴𝑦 ∈ dom 𝐹)) → ((𝐹𝑦) E 𝑥𝑥 ∈ ran 𝐹))
3025, 26, 27, 28, 29syl22anc 839 . . . . . . . . . . . . . 14 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹𝑦) E 𝑥𝑥 ∈ ran 𝐹))
3130ord 865 . . . . . . . . . . . . 13 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (¬ (𝐹𝑦) E 𝑥𝑥 ∈ ran 𝐹))
3224, 31mt3d 148 . . . . . . . . . . . 12 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (𝐹𝑦) E 𝑥)
33 epel 5587 . . . . . . . . . . . 12 ((𝐹𝑦) E 𝑥 ↔ (𝐹𝑦) ∈ 𝑥)
3432, 33sylib 218 . . . . . . . . . . 11 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ 𝑥)
3534ralrimiva 3146 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → ∀𝑦 ∈ dom 𝐹(𝐹𝑦) ∈ 𝑥)
36 ffnfv 7139 . . . . . . . . . 10 (𝐹:dom 𝐹𝑥 ↔ (𝐹 Fn dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹(𝐹𝑦) ∈ 𝑥))
3723, 35, 36sylanbrc 583 . . . . . . . . 9 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹:dom 𝐹𝑥)
389, 22mp1i 13 . . . . . . . . . . . . . 14 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝐹 Fn dom 𝐹)
3917ad2antrr 726 . . . . . . . . . . . . . 14 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → Smo 𝐹)
40 smogt 8407 . . . . . . . . . . . . . 14 ((𝐹 Fn dom 𝐹 ∧ Smo 𝐹𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹𝑦))
4138, 39, 28, 40syl3anc 1373 . . . . . . . . . . . . 13 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹𝑦))
42 ordelon 6408 . . . . . . . . . . . . . . 15 ((Ord dom 𝐹𝑦 ∈ dom 𝐹) → 𝑦 ∈ On)
438, 28, 42sylancr 587 . . . . . . . . . . . . . 14 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ On)
44 simpll 767 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝐴 ⊆ On)
4544, 27sseldd 3984 . . . . . . . . . . . . . 14 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥 ∈ On)
46 ontr2 6431 . . . . . . . . . . . . . 14 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ⊆ (𝐹𝑦) ∧ (𝐹𝑦) ∈ 𝑥) → 𝑦𝑥))
4743, 45, 46syl2anc 584 . . . . . . . . . . . . 13 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝑦 ⊆ (𝐹𝑦) ∧ (𝐹𝑦) ∈ 𝑥) → 𝑦𝑥))
4841, 34, 47mp2and 699 . . . . . . . . . . . 12 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦𝑥)
4948ex 412 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → (𝑦 ∈ dom 𝐹𝑦𝑥))
5049ssrdv 3989 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹𝑥)
5119, 50ssexd 5324 . . . . . . . . 9 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹 ∈ V)
52 fex2 7958 . . . . . . . . 9 ((𝐹:dom 𝐹𝑥 ∧ dom 𝐹 ∈ V ∧ 𝑥𝐴) → 𝐹 ∈ V)
5337, 51, 19, 52syl3anc 1373 . . . . . . . 8 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 ∈ V)
545ordtype2 9574 . . . . . . . 8 (( E We 𝐴 ∧ E Se 𝐴𝐹 ∈ V) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
5520, 21, 53, 54syl3anc 1373 . . . . . . 7 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
56 isof1o 7343 . . . . . . 7 (𝐹 Isom E , E (dom 𝐹, 𝐴) → 𝐹:dom 𝐹1-1-onto𝐴)
57 f1ofo 6855 . . . . . . 7 (𝐹:dom 𝐹1-1-onto𝐴𝐹:dom 𝐹onto𝐴)
58 forn 6823 . . . . . . 7 (𝐹:dom 𝐹onto𝐴 → ran 𝐹 = 𝐴)
5955, 56, 57, 584syl 19 . . . . . 6 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → ran 𝐹 = 𝐴)
6019, 59eleqtrrd 2844 . . . . 5 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥 ∈ ran 𝐹)
6160expr 456 . . . 4 ((𝐴 ⊆ On ∧ 𝑥𝐴) → (¬ 𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹))
6261pm2.18d 127 . . 3 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝑥 ∈ ran 𝐹)
6318, 62eqelssd 4005 . 2 (𝐴 ⊆ On → ran 𝐹 = 𝐴)
6417, 63jca 511 1 (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  wss 3951   class class class wbr 5143   E cep 5583   Se wse 5635   We wwe 5636  dom cdm 5685  ran crn 5686  Ord word 6383  Oncon0 6384   Fn wfn 6556  wf 6557  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561   Isom wiso 6562  Smo wsmo 8385  OrdIsocoi 9549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-smo 8386  df-recs 8411  df-oi 9550
This theorem is referenced by:  oiid  9581  hsmexlem1  10466  hsmexlem2  10467
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