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Theorem oismo 9535
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 5286 (the second statement is trivial under ax-rep 5286). (Contributed by Mario Carneiro, 26-Jun-2015.)
Hypothesis
Ref Expression
oismo.1 𝐹 = OrdIso( E , 𝐴)
Assertion
Ref Expression
oismo (𝐴 βŠ† On β†’ (Smo 𝐹 ∧ ran 𝐹 = 𝐴))
Proof of Theorem oismo
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 epweon 7762 . . . . . 6 E We On
2 wess 5664 . . . . . 6 (𝐴 βŠ† On β†’ ( E We On β†’ E We 𝐴))
31, 2mpi 20 . . . . 5 (𝐴 βŠ† On β†’ E We 𝐴)
4 epse 5660 . . . . 5 E Se 𝐴
5 oismo.1 . . . . . 6 𝐹 = OrdIso( E , 𝐴)
65oiiso2 9526 . . . . 5 (( E We 𝐴 ∧ E Se 𝐴) β†’ 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
73, 4, 6sylancl 587 . . . 4 (𝐴 βŠ† On β†’ 𝐹 Isom E , E (dom 𝐹, ran 𝐹))
85oicl 9524 . . . . 5 Ord dom 𝐹
95oif 9525 . . . . . . 7 𝐹:dom 𝐹⟢𝐴
10 frn 6725 . . . . . . 7 (𝐹:dom 𝐹⟢𝐴 β†’ ran 𝐹 βŠ† 𝐴)
119, 10ax-mp 5 . . . . . 6 ran 𝐹 βŠ† 𝐴
12 id 22 . . . . . 6 (𝐴 βŠ† On β†’ 𝐴 βŠ† On)
1311, 12sstrid 3994 . . . . 5 (𝐴 βŠ† On β†’ ran 𝐹 βŠ† On)
14 smoiso2 8369 . . . . 5 ((Ord dom 𝐹 ∧ ran 𝐹 βŠ† On) β†’ ((𝐹:dom 𝐹–ontoβ†’ran 𝐹 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (dom 𝐹, ran 𝐹)))
158, 13, 14sylancr 588 . . . 4 (𝐴 βŠ† On β†’ ((𝐹:dom 𝐹–ontoβ†’ran 𝐹 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (dom 𝐹, ran 𝐹)))
167, 15mpbird 257 . . 3 (𝐴 βŠ† On β†’ (𝐹:dom 𝐹–ontoβ†’ran 𝐹 ∧ Smo 𝐹))
1716simprd 497 . 2 (𝐴 βŠ† On β†’ Smo 𝐹)
1811a1i 11 . . 3 (𝐴 βŠ† On β†’ ran 𝐹 βŠ† 𝐴)
19 simprl 770 . . . . . 6 ((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) β†’ π‘₯ ∈ 𝐴)
203adantr 482 . . . . . . . 8 ((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) β†’ E We 𝐴)
214a1i 11 . . . . . . . 8 ((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) β†’ E Se 𝐴)
22 ffn 6718 . . . . . . . . . . 11 (𝐹:dom 𝐹⟢𝐴 β†’ 𝐹 Fn dom 𝐹)
239, 22mp1i 13 . . . . . . . . . 10 ((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) β†’ 𝐹 Fn dom 𝐹)
24 simplrr 777 . . . . . . . . . . . . 13 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ Β¬ π‘₯ ∈ ran 𝐹)
253ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ E We 𝐴)
264a1i 11 . . . . . . . . . . . . . . 15 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ E Se 𝐴)
27 simplrl 776 . . . . . . . . . . . . . . 15 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ π‘₯ ∈ 𝐴)
28 simpr 486 . . . . . . . . . . . . . . 15 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ 𝑦 ∈ dom 𝐹)
295oiiniseg 9528 . . . . . . . . . . . . . . 15 ((( E We 𝐴 ∧ E Se 𝐴) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ dom 𝐹)) β†’ ((πΉβ€˜π‘¦) E π‘₯ ∨ π‘₯ ∈ ran 𝐹))
3025, 26, 27, 28, 29syl22anc 838 . . . . . . . . . . . . . 14 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ ((πΉβ€˜π‘¦) E π‘₯ ∨ π‘₯ ∈ ran 𝐹))
3130ord 863 . . . . . . . . . . . . 13 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ (Β¬ (πΉβ€˜π‘¦) E π‘₯ β†’ π‘₯ ∈ ran 𝐹))
3224, 31mt3d 148 . . . . . . . . . . . 12 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ (πΉβ€˜π‘¦) E π‘₯)
33 epel 5584 . . . . . . . . . . . 12 ((πΉβ€˜π‘¦) E π‘₯ ↔ (πΉβ€˜π‘¦) ∈ π‘₯)
3432, 33sylib 217 . . . . . . . . . . 11 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ (πΉβ€˜π‘¦) ∈ π‘₯)
3534ralrimiva 3147 . . . . . . . . . 10 ((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) β†’ βˆ€π‘¦ ∈ dom 𝐹(πΉβ€˜π‘¦) ∈ π‘₯)
36 ffnfv 7118 . . . . . . . . . 10 (𝐹:dom 𝐹⟢π‘₯ ↔ (𝐹 Fn dom 𝐹 ∧ βˆ€π‘¦ ∈ dom 𝐹(πΉβ€˜π‘¦) ∈ π‘₯))
3723, 35, 36sylanbrc 584 . . . . . . . . 9 ((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) β†’ 𝐹:dom 𝐹⟢π‘₯)
389, 22mp1i 13 . . . . . . . . . . . . . 14 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ 𝐹 Fn dom 𝐹)
3917ad2antrr 725 . . . . . . . . . . . . . 14 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ Smo 𝐹)
40 smogt 8367 . . . . . . . . . . . . . 14 ((𝐹 Fn dom 𝐹 ∧ Smo 𝐹 ∧ 𝑦 ∈ dom 𝐹) β†’ 𝑦 βŠ† (πΉβ€˜π‘¦))
4138, 39, 28, 40syl3anc 1372 . . . . . . . . . . . . 13 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ 𝑦 βŠ† (πΉβ€˜π‘¦))
42 ordelon 6389 . . . . . . . . . . . . . . 15 ((Ord dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) β†’ 𝑦 ∈ On)
438, 28, 42sylancr 588 . . . . . . . . . . . . . 14 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ 𝑦 ∈ On)
44 simpll 766 . . . . . . . . . . . . . . 15 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ 𝐴 βŠ† On)
4544, 27sseldd 3984 . . . . . . . . . . . . . 14 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ π‘₯ ∈ On)
46 ontr2 6412 . . . . . . . . . . . . . 14 ((𝑦 ∈ On ∧ π‘₯ ∈ On) β†’ ((𝑦 βŠ† (πΉβ€˜π‘¦) ∧ (πΉβ€˜π‘¦) ∈ π‘₯) β†’ 𝑦 ∈ π‘₯))
4743, 45, 46syl2anc 585 . . . . . . . . . . . . 13 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ ((𝑦 βŠ† (πΉβ€˜π‘¦) ∧ (πΉβ€˜π‘¦) ∈ π‘₯) β†’ 𝑦 ∈ π‘₯))
4841, 34, 47mp2and 698 . . . . . . . . . . . 12 (((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) β†’ 𝑦 ∈ π‘₯)
4948ex 414 . . . . . . . . . . 11 ((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) β†’ (𝑦 ∈ dom 𝐹 β†’ 𝑦 ∈ π‘₯))
5049ssrdv 3989 . . . . . . . . . 10 ((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) β†’ dom 𝐹 βŠ† π‘₯)
5119, 50ssexd 5325 . . . . . . . . 9 ((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) β†’ dom 𝐹 ∈ V)
52 fex2 7924 . . . . . . . . 9 ((𝐹:dom 𝐹⟢π‘₯ ∧ dom 𝐹 ∈ V ∧ π‘₯ ∈ 𝐴) β†’ 𝐹 ∈ V)
5337, 51, 19, 52syl3anc 1372 . . . . . . . 8 ((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) β†’ 𝐹 ∈ V)
545ordtype2 9529 . . . . . . . 8 (( E We 𝐴 ∧ E Se 𝐴 ∧ 𝐹 ∈ V) β†’ 𝐹 Isom E , E (dom 𝐹, 𝐴))
5520, 21, 53, 54syl3anc 1372 . . . . . . 7 ((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) β†’ 𝐹 Isom E , E (dom 𝐹, 𝐴))
56 isof1o 7320 . . . . . . 7 (𝐹 Isom E , E (dom 𝐹, 𝐴) β†’ 𝐹:dom 𝐹–1-1-onto→𝐴)
57 f1ofo 6841 . . . . . . 7 (𝐹:dom 𝐹–1-1-onto→𝐴 β†’ 𝐹:dom 𝐹–onto→𝐴)
58 forn 6809 . . . . . . 7 (𝐹:dom 𝐹–onto→𝐴 β†’ ran 𝐹 = 𝐴)
5955, 56, 57, 584syl 19 . . . . . 6 ((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) β†’ ran 𝐹 = 𝐴)
6019, 59eleqtrrd 2837 . . . . 5 ((𝐴 βŠ† On ∧ (π‘₯ ∈ 𝐴 ∧ Β¬ π‘₯ ∈ ran 𝐹)) β†’ π‘₯ ∈ ran 𝐹)
6160expr 458 . . . 4 ((𝐴 βŠ† On ∧ π‘₯ ∈ 𝐴) β†’ (Β¬ π‘₯ ∈ ran 𝐹 β†’ π‘₯ ∈ ran 𝐹))
6261pm2.18d 127 . . 3 ((𝐴 βŠ† On ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ ran 𝐹)
6318, 62eqelssd 4004 . 2 (𝐴 βŠ† On β†’ ran 𝐹 = 𝐴)
6417, 63jca 513 1 (𝐴 βŠ† On β†’ (Smo 𝐹 ∧ ran 𝐹 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475   βŠ† wss 3949   class class class wbr 5149   E cep 5580   Se wse 5630   We wwe 5631  dom cdm 5677  ran crn 5678  Ord word 6364  Oncon0 6365   Fn wfn 6539  βŸΆwf 6540  β€“ontoβ†’wfo 6542  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544   Isom wiso 6545  Smo wsmo 8345  OrdIsocoi 9504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-smo 8346  df-recs 8371  df-oi 9505
This theorem is referenced by:  oiid  9536  hsmexlem1  10421  hsmexlem2  10422
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