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Theorem hsmexlem1 10423
Description: Lemma for hsmex 10429. Bound the order type of a limited-cardinality set of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypothesis
Ref Expression
hsmexlem.o 𝑂 = OrdIso( E , 𝐴)
Assertion
Ref Expression
hsmexlem1 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ dom 𝑂 ∈ (harβ€˜π’« 𝐡))
Proof of Theorem hsmexlem1
StepHypRef Expression
1 hsmexlem.o . . . 4 𝑂 = OrdIso( E , 𝐴)
21oicl 9526 . . 3 Ord dom 𝑂
3 relwdom 9563 . . . . . . . 8 Rel β‰Ό*
43brrelex1i 5732 . . . . . . 7 (𝐴 β‰Ό* 𝐡 β†’ 𝐴 ∈ V)
54adantl 482 . . . . . 6 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ 𝐴 ∈ V)
6 uniexg 7732 . . . . . 6 (𝐴 ∈ V β†’ βˆͺ 𝐴 ∈ V)
7 sucexg 7795 . . . . . 6 (βˆͺ 𝐴 ∈ V β†’ suc βˆͺ 𝐴 ∈ V)
85, 6, 73syl 18 . . . . 5 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ suc βˆͺ 𝐴 ∈ V)
91oif 9527 . . . . . . 7 𝑂:dom π‘‚βŸΆπ΄
10 onsucuni 7818 . . . . . . . 8 (𝐴 βŠ† On β†’ 𝐴 βŠ† suc βˆͺ 𝐴)
1110adantr 481 . . . . . . 7 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ 𝐴 βŠ† suc βˆͺ 𝐴)
12 fss 6734 . . . . . . 7 ((𝑂:dom π‘‚βŸΆπ΄ ∧ 𝐴 βŠ† suc βˆͺ 𝐴) β†’ 𝑂:dom π‘‚βŸΆsuc βˆͺ 𝐴)
139, 11, 12sylancr 587 . . . . . 6 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ 𝑂:dom π‘‚βŸΆsuc βˆͺ 𝐴)
141oismo 9537 . . . . . . . 8 (𝐴 βŠ† On β†’ (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
1514adantr 481 . . . . . . 7 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
1615simpld 495 . . . . . 6 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ Smo 𝑂)
17 ssorduni 7768 . . . . . . . 8 (𝐴 βŠ† On β†’ Ord βˆͺ 𝐴)
1817adantr 481 . . . . . . 7 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ Ord βˆͺ 𝐴)
19 ordsuc 7803 . . . . . . 7 (Ord βˆͺ 𝐴 ↔ Ord suc βˆͺ 𝐴)
2018, 19sylib 217 . . . . . 6 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ Ord suc βˆͺ 𝐴)
21 smocdmdom 8370 . . . . . 6 ((𝑂:dom π‘‚βŸΆsuc βˆͺ 𝐴 ∧ Smo 𝑂 ∧ Ord suc βˆͺ 𝐴) β†’ dom 𝑂 βŠ† suc βˆͺ 𝐴)
2213, 16, 20, 21syl3anc 1371 . . . . 5 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ dom 𝑂 βŠ† suc βˆͺ 𝐴)
238, 22ssexd 5324 . . . 4 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ dom 𝑂 ∈ V)
24 elong 6372 . . . 4 (dom 𝑂 ∈ V β†’ (dom 𝑂 ∈ On ↔ Ord dom 𝑂))
2523, 24syl 17 . . 3 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ (dom 𝑂 ∈ On ↔ Ord dom 𝑂))
262, 25mpbiri 257 . 2 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ dom 𝑂 ∈ On)
27 canth2g 9133 . . . 4 (dom 𝑂 ∈ V β†’ dom 𝑂 β‰Ί 𝒫 dom 𝑂)
28 sdomdom 8978 . . . 4 (dom 𝑂 β‰Ί 𝒫 dom 𝑂 β†’ dom 𝑂 β‰Ό 𝒫 dom 𝑂)
2923, 27, 283syl 18 . . 3 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ dom 𝑂 β‰Ό 𝒫 dom 𝑂)
30 simpl 483 . . . . . . . . . . 11 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ 𝐴 βŠ† On)
31 epweon 7764 . . . . . . . . . . 11 E We On
32 wess 5663 . . . . . . . . . . 11 (𝐴 βŠ† On β†’ ( E We On β†’ E We 𝐴))
3330, 31, 32mpisyl 21 . . . . . . . . . 10 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ E We 𝐴)
34 epse 5659 . . . . . . . . . 10 E Se 𝐴
351oiiso2 9528 . . . . . . . . . 10 (( E We 𝐴 ∧ E Se 𝐴) β†’ 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
3633, 34, 35sylancl 586 . . . . . . . . 9 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
37 isof1o 7322 . . . . . . . . 9 (𝑂 Isom E , E (dom 𝑂, ran 𝑂) β†’ 𝑂:dom 𝑂–1-1-ontoβ†’ran 𝑂)
3836, 37syl 17 . . . . . . . 8 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ 𝑂:dom 𝑂–1-1-ontoβ†’ran 𝑂)
3915simprd 496 . . . . . . . . 9 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ ran 𝑂 = 𝐴)
4039f1oeq3d 6830 . . . . . . . 8 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ (𝑂:dom 𝑂–1-1-ontoβ†’ran 𝑂 ↔ 𝑂:dom 𝑂–1-1-onto→𝐴))
4138, 40mpbid 231 . . . . . . 7 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ 𝑂:dom 𝑂–1-1-onto→𝐴)
42 f1oen2g 8966 . . . . . . 7 ((dom 𝑂 ∈ On ∧ 𝐴 ∈ V ∧ 𝑂:dom 𝑂–1-1-onto→𝐴) β†’ dom 𝑂 β‰ˆ 𝐴)
4326, 5, 41, 42syl3anc 1371 . . . . . 6 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ dom 𝑂 β‰ˆ 𝐴)
44 endom 8977 . . . . . 6 (dom 𝑂 β‰ˆ 𝐴 β†’ dom 𝑂 β‰Ό 𝐴)
45 domwdom 9571 . . . . . 6 (dom 𝑂 β‰Ό 𝐴 β†’ dom 𝑂 β‰Ό* 𝐴)
4643, 44, 453syl 18 . . . . 5 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ dom 𝑂 β‰Ό* 𝐴)
47 wdomtr 9572 . . . . 5 ((dom 𝑂 β‰Ό* 𝐴 ∧ 𝐴 β‰Ό* 𝐡) β†’ dom 𝑂 β‰Ό* 𝐡)
4846, 47sylancom 588 . . . 4 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ dom 𝑂 β‰Ό* 𝐡)
49 wdompwdom 9575 . . . 4 (dom 𝑂 β‰Ό* 𝐡 β†’ 𝒫 dom 𝑂 β‰Ό 𝒫 𝐡)
5048, 49syl 17 . . 3 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ 𝒫 dom 𝑂 β‰Ό 𝒫 𝐡)
51 domtr 9005 . . 3 ((dom 𝑂 β‰Ό 𝒫 dom 𝑂 ∧ 𝒫 dom 𝑂 β‰Ό 𝒫 𝐡) β†’ dom 𝑂 β‰Ό 𝒫 𝐡)
5229, 50, 51syl2anc 584 . 2 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ dom 𝑂 β‰Ό 𝒫 𝐡)
53 elharval 9558 . 2 (dom 𝑂 ∈ (harβ€˜π’« 𝐡) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 β‰Ό 𝒫 𝐡))
5426, 52, 53sylanbrc 583 1 ((𝐴 βŠ† On ∧ 𝐴 β‰Ό* 𝐡) β†’ dom 𝑂 ∈ (harβ€˜π’« 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908   class class class wbr 5148   E cep 5579   Se wse 5629   We wwe 5630  dom cdm 5676  ran crn 5677  Ord word 6363  Oncon0 6364  suc csuc 6366  βŸΆwf 6539  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543   Isom wiso 6544  Smo wsmo 8347   β‰ˆ cen 8938   β‰Ό cdom 8939   β‰Ί csdm 8940  OrdIsocoi 9506  harchar 9553   β‰Ό* cwdom 9561
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-smo 8348  df-recs 8373  df-en 8942  df-dom 8943  df-sdom 8944  df-oi 9507  df-har 9554  df-wdom 9562
This theorem is referenced by:  hsmexlem2  10424  hsmexlem4  10426
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