MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oismo Structured version   Visualization version   GIF version

Theorem oismo 8857
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 5088 (the second statement is trivial under ax-rep 5088). (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 7360 . . . . . 6 E We On
2 wess 5437 . . . . . 6 (𝐴 ⊆ On → ( E We On → E We 𝐴))
31, 2mpi 20 . . . . 5 (𝐴 ⊆ On → E We 𝐴)
4 epse 5433 . . . . 5 E Se 𝐴
5 oismo.1 . . . . . 6 𝐹 = OrdIso( E , 𝐴)
65oiiso2 8848 . . . . 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 8846 . . . . 5 Ord dom 𝐹
95oif 8847 . . . . . . 7 𝐹:dom 𝐹𝐴
10 frn 6395 . . . . . . 7 (𝐹:dom 𝐹𝐴 → ran 𝐹𝐴)
119, 10ax-mp 5 . . . . . 6 ran 𝐹𝐴
12 id 22 . . . . . 6 (𝐴 ⊆ On → 𝐴 ⊆ On)
1311, 12sstrid 3906 . . . . 5 (𝐴 ⊆ On → ran 𝐹 ⊆ On)
14 smoiso2 7865 . . . . 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 258 . . 3 (𝐴 ⊆ On → (𝐹:dom 𝐹onto→ran 𝐹 ∧ Smo 𝐹))
1716simprd 496 . 2 (𝐴 ⊆ On → Smo 𝐹)
1811a1i 11 . . 3 (𝐴 ⊆ On → ran 𝐹𝐴)
19 simprl 767 . . . . . 6 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥𝐴)
203adantr 481 . . . . . . . 8 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → E We 𝐴)
214a1i 11 . . . . . . . 8 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → E Se 𝐴)
22 ffn 6389 . . . . . . . . . . 11 (𝐹:dom 𝐹𝐴𝐹 Fn dom 𝐹)
239, 22mp1i 13 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Fn dom 𝐹)
24 simplrr 774 . . . . . . . . . . . . 13 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
253ad2antrr 722 . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥𝐴)
28 simpr 485 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
295oiiniseg 8850 . . . . . . . . . . . . . . 15 ((( E We 𝐴 ∧ E Se 𝐴) ∧ (𝑥𝐴𝑦 ∈ dom 𝐹)) → ((𝐹𝑦) E 𝑥𝑥 ∈ ran 𝐹))
3025, 26, 27, 28, 29syl22anc 835 . . . . . . . . . . . . . 14 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹𝑦) E 𝑥𝑥 ∈ ran 𝐹))
3130ord 859 . . . . . . . . . . . . 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 5364 . . . . . . . . . . . 12 ((𝐹𝑦) E 𝑥 ↔ (𝐹𝑦) ∈ 𝑥)
3432, 33sylib 219 . . . . . . . . . . 11 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ 𝑥)
3534ralrimiva 3151 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → ∀𝑦 ∈ dom 𝐹(𝐹𝑦) ∈ 𝑥)
36 ffnfv 6752 . . . . . . . . . 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 722 . . . . . . . . . . . . . 14 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → Smo 𝐹)
40 smogt 7863 . . . . . . . . . . . . . 14 ((𝐹 Fn dom 𝐹 ∧ Smo 𝐹𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹𝑦))
4138, 39, 28, 40syl3anc 1364 . . . . . . . . . . . . 13 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹𝑦))
42 ordelon 6097 . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝐴 ⊆ On)
4544, 27sseldd 3896 . . . . . . . . . . . . . 14 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥 ∈ On)
46 ontr2 6120 . . . . . . . . . . . . . 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 695 . . . . . . . . . . . 12 (((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦𝑥)
4948ex 413 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → (𝑦 ∈ dom 𝐹𝑦𝑥))
5049ssrdv 3901 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹𝑥)
5119, 50ssexd 5126 . . . . . . . . 9 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹 ∈ V)
52 fex2 7501 . . . . . . . . 9 ((𝐹:dom 𝐹𝑥 ∧ dom 𝐹 ∈ V ∧ 𝑥𝐴) → 𝐹 ∈ V)
5337, 51, 19, 52syl3anc 1364 . . . . . . . 8 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 ∈ V)
545ordtype2 8851 . . . . . . . 8 (( E We 𝐴 ∧ E Se 𝐴𝐹 ∈ V) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
5520, 21, 53, 54syl3anc 1364 . . . . . . 7 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Isom E , E (dom 𝐹, 𝐴))
56 isof1o 6946 . . . . . . 7 (𝐹 Isom E , E (dom 𝐹, 𝐴) → 𝐹:dom 𝐹1-1-onto𝐴)
57 f1ofo 6497 . . . . . . 7 (𝐹:dom 𝐹1-1-onto𝐴𝐹:dom 𝐹onto𝐴)
58 forn 6468 . . . . . . 7 (𝐹:dom 𝐹onto𝐴 → ran 𝐹 = 𝐴)
5955, 56, 57, 584syl 19 . . . . . 6 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → ran 𝐹 = 𝐴)
6019, 59eleqtrrd 2888 . . . . 5 ((𝐴 ⊆ On ∧ (𝑥𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥 ∈ ran 𝐹)
6160expr 457 . . . 4 ((𝐴 ⊆ On ∧ 𝑥𝐴) → (¬ 𝑥 ∈ ran 𝐹𝑥 ∈ ran 𝐹))
6261pm2.18d 127 . . 3 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝑥 ∈ ran 𝐹)
6318, 62eqelssd 3915 . 2 (𝐴 ⊆ On → ran 𝐹 = 𝐴)
6417, 63jca 512 1 (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 842   = wceq 1525  wcel 2083  wral 3107  Vcvv 3440  wss 3865   class class class wbr 4968   E cep 5359   Se wse 5407   We wwe 5408  dom cdm 5450  ran crn 5451  Ord word 6072  Oncon0 6073   Fn wfn 6227  wf 6228  ontowfo 6230  1-1-ontowf1o 6231  cfv 6232   Isom wiso 6233  Smo wsmo 7841  OrdIsocoi 8826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-wrecs 7805  df-smo 7842  df-recs 7867  df-oi 8827
This theorem is referenced by:  oiid  8858  hsmexlem1  9701  hsmexlem2  9702
  Copyright terms: Public domain W3C validator