MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hsmexlem2 Structured version   Visualization version   GIF version

Theorem hsmexlem2 9837
Description: Lemma for hsmex 9842. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9985 but use of order types allows to canonically choose the sub-bijections, removing the choice requirement. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by AV, 18-Sep-2021.)
Hypotheses
Ref Expression
hsmexlem.f 𝐹 = OrdIso( E , 𝐵)
hsmexlem.g 𝐺 = OrdIso( E , 𝑎𝐴 𝐵)
Assertion
Ref Expression
hsmexlem2 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
Distinct variable groups:   𝐴,𝑎   𝐶,𝑎
Allowed substitution hints:   𝐵(𝑎)   𝐹(𝑎)   𝐺(𝑎)   𝑉(𝑎)

Proof of Theorem hsmexlem2
Dummy variables 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpwi 4547 . . . . . 6 (𝐵 ∈ 𝒫 On → 𝐵 ⊆ On)
21adantr 481 . . . . 5 ((𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → 𝐵 ⊆ On)
32ralimi 3157 . . . 4 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → ∀𝑎𝐴 𝐵 ⊆ On)
4 iunss 4960 . . . 4 ( 𝑎𝐴 𝐵 ⊆ On ↔ ∀𝑎𝐴 𝐵 ⊆ On)
53, 4sylibr 235 . . 3 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → 𝑎𝐴 𝐵 ⊆ On)
653ad2ant3 1127 . 2 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝑎𝐴 𝐵 ⊆ On)
7 xpexg 7462 . . . 4 ((𝐴𝑉𝐶 ∈ On) → (𝐴 × 𝐶) ∈ V)
873adant3 1124 . . 3 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝐴 × 𝐶) ∈ V)
9 nfv 1906 . . . . . . . . 9 𝑎 𝐶 ∈ On
10 nfra1 3216 . . . . . . . . 9 𝑎𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)
119, 10nfan 1891 . . . . . . . 8 𝑎(𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶))
12 rsp 3202 . . . . . . . . 9 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → (𝑎𝐴 → (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)))
13 onelss 6226 . . . . . . . . . . . . . 14 (𝐶 ∈ On → (dom 𝐹𝐶 → dom 𝐹𝐶))
1413imp 407 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ dom 𝐹𝐶) → dom 𝐹𝐶)
1514adantrl 712 . . . . . . . . . . . 12 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐹𝐶)
16153adant3 1124 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) ∧ 𝑏𝐵) → dom 𝐹𝐶)
17 hsmexlem.f . . . . . . . . . . . . . . . . . . 19 𝐹 = OrdIso( E , 𝐵)
1817oismo 8992 . . . . . . . . . . . . . . . . . 18 (𝐵 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
191, 18syl 17 . . . . . . . . . . . . . . . . 17 (𝐵 ∈ 𝒫 On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
2019ad2antrl 724 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
2120simprd 496 . . . . . . . . . . . . . . 15 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → ran 𝐹 = 𝐵)
2217oif 8982 . . . . . . . . . . . . . . 15 𝐹:dom 𝐹𝐵
2321, 22jctil 520 . . . . . . . . . . . . . 14 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝐹:dom 𝐹𝐵 ∧ ran 𝐹 = 𝐵))
24 dffo2 6587 . . . . . . . . . . . . . 14 (𝐹:dom 𝐹onto𝐵 ↔ (𝐹:dom 𝐹𝐵 ∧ ran 𝐹 = 𝐵))
2523, 24sylibr 235 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝐹:dom 𝐹onto𝐵)
26 dffo3 6860 . . . . . . . . . . . . . 14 (𝐹:dom 𝐹onto𝐵 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑏𝐵𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒)))
2726simprbi 497 . . . . . . . . . . . . 13 (𝐹:dom 𝐹onto𝐵 → ∀𝑏𝐵𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒))
28 rsp 3202 . . . . . . . . . . . . 13 (∀𝑏𝐵𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒) → (𝑏𝐵 → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒)))
2925, 27, 283syl 18 . . . . . . . . . . . 12 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝑏𝐵 → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒)))
30293impia 1109 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) ∧ 𝑏𝐵) → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒))
31 ssrexv 4031 . . . . . . . . . . 11 (dom 𝐹𝐶 → (∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒) → ∃𝑒𝐶 𝑏 = (𝐹𝑒)))
3216, 30, 31sylc 65 . . . . . . . . . 10 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) ∧ 𝑏𝐵) → ∃𝑒𝐶 𝑏 = (𝐹𝑒))
33323exp 1111 . . . . . . . . 9 (𝐶 ∈ On → ((𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → (𝑏𝐵 → ∃𝑒𝐶 𝑏 = (𝐹𝑒))))
3412, 33sylan9r 509 . . . . . . . 8 ((𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝑎𝐴 → (𝑏𝐵 → ∃𝑒𝐶 𝑏 = (𝐹𝑒))))
3511, 34reximdai 3308 . . . . . . 7 ((𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒)))
36353adant1 1122 . . . . . 6 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒)))
37 nfv 1906 . . . . . . 7 𝑑𝑒𝐶 𝑏 = (𝐹𝑒)
38 nfcv 2974 . . . . . . . 8 𝑎𝐶
39 nfcv 2974 . . . . . . . . . . 11 𝑎 E
40 nfcsb1v 3904 . . . . . . . . . . 11 𝑎𝑑 / 𝑎𝐵
4139, 40nfoi 8966 . . . . . . . . . 10 𝑎OrdIso( E , 𝑑 / 𝑎𝐵)
42 nfcv 2974 . . . . . . . . . 10 𝑎𝑒
4341, 42nffv 6673 . . . . . . . . 9 𝑎(OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
4443nfeq2 2992 . . . . . . . 8 𝑎 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
4538, 44nfrex 3306 . . . . . . 7 𝑎𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
46 csbeq1a 3894 . . . . . . . . . . . 12 (𝑎 = 𝑑𝐵 = 𝑑 / 𝑎𝐵)
47 oieq2 8965 . . . . . . . . . . . 12 (𝐵 = 𝑑 / 𝑎𝐵 → OrdIso( E , 𝐵) = OrdIso( E , 𝑑 / 𝑎𝐵))
4846, 47syl 17 . . . . . . . . . . 11 (𝑎 = 𝑑 → OrdIso( E , 𝐵) = OrdIso( E , 𝑑 / 𝑎𝐵))
4917, 48syl5eq 2865 . . . . . . . . . 10 (𝑎 = 𝑑𝐹 = OrdIso( E , 𝑑 / 𝑎𝐵))
5049fveq1d 6665 . . . . . . . . 9 (𝑎 = 𝑑 → (𝐹𝑒) = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
5150eqeq2d 2829 . . . . . . . 8 (𝑎 = 𝑑 → (𝑏 = (𝐹𝑒) ↔ 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
5251rexbidv 3294 . . . . . . 7 (𝑎 = 𝑑 → (∃𝑒𝐶 𝑏 = (𝐹𝑒) ↔ ∃𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
5337, 45, 52cbvrexw 3440 . . . . . 6 (∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒) ↔ ∃𝑑𝐴𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
5436, 53syl6ib 252 . . . . 5 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑑𝐴𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
55 eliun 4914 . . . . 5 (𝑏 𝑎𝐴 𝐵 ↔ ∃𝑎𝐴 𝑏𝐵)
56 vex 3495 . . . . . . . . . . 11 𝑑 ∈ V
57 vex 3495 . . . . . . . . . . 11 𝑒 ∈ V
5856, 57op1std 7688 . . . . . . . . . 10 (𝑐 = ⟨𝑑, 𝑒⟩ → (1st𝑐) = 𝑑)
5958csbeq1d 3884 . . . . . . . . 9 (𝑐 = ⟨𝑑, 𝑒⟩ → (1st𝑐) / 𝑎𝐵 = 𝑑 / 𝑎𝐵)
60 oieq2 8965 . . . . . . . . 9 ((1st𝑐) / 𝑎𝐵 = 𝑑 / 𝑎𝐵 → OrdIso( E , (1st𝑐) / 𝑎𝐵) = OrdIso( E , 𝑑 / 𝑎𝐵))
6159, 60syl 17 . . . . . . . 8 (𝑐 = ⟨𝑑, 𝑒⟩ → OrdIso( E , (1st𝑐) / 𝑎𝐵) = OrdIso( E , 𝑑 / 𝑎𝐵))
6256, 57op2ndd 7689 . . . . . . . 8 (𝑐 = ⟨𝑑, 𝑒⟩ → (2nd𝑐) = 𝑒)
6361, 62fveq12d 6670 . . . . . . 7 (𝑐 = ⟨𝑑, 𝑒⟩ → (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
6463eqeq2d 2829 . . . . . 6 (𝑐 = ⟨𝑑, 𝑒⟩ → (𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) ↔ 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
6564rexxp 5706 . . . . 5 (∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) ↔ ∃𝑑𝐴𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
6654, 55, 653imtr4g 297 . . . 4 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝑏 𝑎𝐴 𝐵 → ∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐))))
6766imp 407 . . 3 (((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) ∧ 𝑏 𝑎𝐴 𝐵) → ∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)))
688, 67wdomd 9033 . 2 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝑎𝐴 𝐵* (𝐴 × 𝐶))
69 hsmexlem.g . . 3 𝐺 = OrdIso( E , 𝑎𝐴 𝐵)
7069hsmexlem1 9836 . 2 (( 𝑎𝐴 𝐵 ⊆ On ∧ 𝑎𝐴 𝐵* (𝐴 × 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
716, 68, 70syl2anc 584 1 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  csb 3880  wss 3933  𝒫 cpw 4535  cop 4563   ciun 4910   class class class wbr 5057   E cep 5457   × cxp 5546  dom cdm 5548  ran crn 5549  Oncon0 6184  wf 6344  ontowfo 6346  cfv 6348  1st c1st 7676  2nd c2nd 7677  Smo wsmo 7971  OrdIsocoi 8961  harchar 9008  * cwdom 9009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-smo 7972  df-recs 7997  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-oi 8962  df-har 9010  df-wdom 9011
This theorem is referenced by:  hsmexlem3  9838
  Copyright terms: Public domain W3C validator