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Theorem oismo 8990
 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 5154 (the second statement is trivial under ax-rep 5154). (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 7479 . . . . . 6 E We On
2 wess 5506 . . . . . 6 (𝐴 ⊆ On → ( E We On → E We 𝐴))
31, 2mpi 20 . . . . 5 (𝐴 ⊆ On → E We 𝐴)
4 epse 5502 . . . . 5 E Se 𝐴
5 oismo.1 . . . . . 6 𝐹 = OrdIso( E , 𝐴)
65oiiso2 8981 . . . . 5 (( E We 𝐴 ∧ E Se 𝐴) → 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
73, 4, 6sylancl 589 . . . 4 (𝐴 ⊆ On → 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
85oicl 8979 . . . . 5 Ord dom 𝐹
95oif 8980 . . . . . . 7 𝐹:dom 𝐹𝐴
10 frn 6493 . . . . . . 7 (𝐹:dom 𝐹𝐴 → ran 𝐹𝐴)
119, 10ax-mp 5 . . . . . 6 ran 𝐹𝐴
12 id 22 . . . . . 6 (𝐴 ⊆ On → 𝐴 ⊆ On)
1311, 12sstrid 3926 . . . . 5 (𝐴 ⊆ On → ran 𝐹 ⊆ On)
14 smoiso2 7991 . . . . 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 590 . . . 4 (𝐴 ⊆ On → ((𝐹:dom 𝐹onto→ran 𝐹 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (dom 𝐹, ran 𝐹)))
167, 15mpbird 260 . . 3 (𝐴 ⊆ On → (𝐹:dom 𝐹onto→ran 𝐹 ∧ Smo 𝐹))
1716simprd 499 . 2 (𝐴 ⊆ On → Smo 𝐹)
1811a1i 11 . . 3 (𝐴 ⊆ On → ran 𝐹𝐴)
19 simprl 770 . . . . . 6 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥𝐴)
203adantr 484 . . . . . . . 8 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → E We 𝐴)
214a1i 11 . . . . . . . 8 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → E Se 𝐴)
22 ffn 6487 . . . . . . . . . . 11 (𝐹:dom 𝐹𝐴𝐹 Fn dom 𝐹)
239, 22mp1i 13 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Fn dom 𝐹)
24 simplrr 777 . . . . . . . . . . . . 13 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
253ad2antrr 725 . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥𝐴)
28 simpr 488 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
295oiiniseg 8983 . . . . . . . . . . . . . . 15 ((( E We 𝐴 ∧ E Se 𝐴) ∧ (𝑥𝐴𝑦 ∈ dom 𝐹)) → ((𝐹𝑦) E 𝑥𝑥 ∈ ran 𝐹))
3025, 26, 27, 28, 29syl22anc 837 . . . . . . . . . . . . . 14 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹𝑦) E 𝑥𝑥 ∈ ran 𝐹))
3130ord 861 . . . . . . . . . . . . 13 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (¬ (𝐹𝑦) E 𝑥𝑥 ∈ ran 𝐹))
3224, 31mt3d 150 . . . . . . . . . . . 12 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (𝐹𝑦) E 𝑥)
33 epel 5433 . . . . . . . . . . . 12 ((𝐹𝑦) E 𝑥 ↔ (𝐹𝑦) ∈ 𝑥)
3432, 33sylib 221 . . . . . . . . . . 11 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ 𝑥)
3534ralrimiva 3149 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → ∀𝑦 ∈ dom 𝐹(𝐹𝑦) ∈ 𝑥)
36 ffnfv 6859 . . . . . . . . . 10 (𝐹:dom 𝐹𝑥 ↔ (𝐹 Fn dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹(𝐹𝑦) ∈ 𝑥))
3723, 35, 36sylanbrc 586 . . . . . . . . 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 725 . . . . . . . . . . . . . 14 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → Smo 𝐹)
40 smogt 7989 . . . . . . . . . . . . . 14 ((𝐹 Fn dom 𝐹 ∧ Smo 𝐹𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹𝑦))
4138, 39, 28, 40syl3anc 1368 . . . . . . . . . . . . 13 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹𝑦))
42 ordelon 6183 . . . . . . . . . . . . . . 15 ((Ord dom 𝐹𝑦 ∈ dom 𝐹) → 𝑦 ∈ On)
438, 28, 42sylancr 590 . . . . . . . . . . . . . 14 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ On)
44 simpll 766 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝐴 ⊆ On)
4544, 27sseldd 3916 . . . . . . . . . . . . . 14 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥 ∈ On)
46 ontr2 6206 . . . . . . . . . . . . . 14 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ⊆ (𝐹𝑦) ∧ (𝐹𝑦) ∈ 𝑥) → 𝑦𝑥))
4743, 45, 46syl2anc 587 . . . . . . . . . . . . 13 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝑦 ⊆ (𝐹𝑦) ∧ (𝐹𝑦) ∈ 𝑥) → 𝑦𝑥))
4841, 34, 47mp2and 698 . . . . . . . . . . . 12 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦𝑥)
4948ex 416 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → (𝑦 ∈ dom 𝐹𝑦𝑥))
5049ssrdv 3921 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹𝑥)
5119, 50ssexd 5192 . . . . . . . . 9 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹 ∈ V)
52 fex2 7622 . . . . . . . . 9 ((𝐹:dom 𝐹𝑥 ∧ dom 𝐹 ∈ V ∧ 𝑥𝐴) → 𝐹 ∈ V)
5337, 51, 19, 52syl3anc 1368 . . . . . . . 8 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 ∈ V)
545ordtype2 8984 . . . . . . . 8 (( E We 𝐴 ∧ E Se 𝐴𝐹 ∈ V) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
5520, 21, 53, 54syl3anc 1368 . . . . . . 7 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
56 isof1o 7055 . . . . . . 7 (𝐹 Isom E , E (dom 𝐹, 𝐴) → 𝐹:dom 𝐹1-1-onto𝐴)
57 f1ofo 6597 . . . . . . 7 (𝐹:dom 𝐹1-1-onto𝐴𝐹:dom 𝐹onto𝐴)
58 forn 6568 . . . . . . 7 (𝐹:dom 𝐹onto𝐴 → ran 𝐹 = 𝐴)
5955, 56, 57, 584syl 19 . . . . . 6 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → ran 𝐹 = 𝐴)
6019, 59eleqtrrd 2893 . . . . 5 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥 ∈ ran 𝐹)
6160expr 460 . . . 4 ((𝐴 ⊆ On ∧ 𝑥𝐴) → (¬ 𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹))
6261pm2.18d 127 . . 3 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝑥 ∈ ran 𝐹)
6318, 62eqelssd 3936 . 2 (𝐴 ⊆ On → ran 𝐹 = 𝐴)
6417, 63jca 515 1 (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030   E cep 5429   Se wse 5476   We wwe 5477  dom cdm 5519  ran crn 5520  Ord word 6158  Oncon0 6159   Fn wfn 6319  ⟶wf 6320  –onto→wfo 6322  –1-1-onto→wf1o 6323  ‘cfv 6324   Isom wiso 6325  Smo wsmo 7967  OrdIsocoi 8959 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-wrecs 7932  df-smo 7968  df-recs 7993  df-oi 8960 This theorem is referenced by:  oiid  8991  hsmexlem1  9839  hsmexlem2  9840
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