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Theorem unwdomg 8698
Description: Weak dominance of a (disjoint) union. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
unwdomg ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≼* (𝐵𝐷))

Proof of Theorem unwdomg
Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brwdom3i 8697 . . 3 (𝐴* 𝐵 → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
213ad2ant1 1163 . 2 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
3 brwdom3i 8697 . . . . 5 (𝐶* 𝐷 → ∃𝑔𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))
433ad2ant2 1164 . . . 4 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → ∃𝑔𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))
54adantr 472 . . 3 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → ∃𝑔𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))
6 relwdom 8680 . . . . . . . . . 10 Rel ≼*
76brrelex1i 5330 . . . . . . . . 9 (𝐴* 𝐵𝐴 ∈ V)
86brrelex1i 5330 . . . . . . . . 9 (𝐶* 𝐷𝐶 ∈ V)
9 unexg 7159 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴𝐶) ∈ V)
107, 8, 9syl2an 589 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐴𝐶) ∈ V)
11103adant3 1162 . . . . . . 7 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ∈ V)
1211adantr 472 . . . . . 6 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) → (𝐴𝐶) ∈ V)
136brrelex2i 5331 . . . . . . . . 9 (𝐴* 𝐵𝐵 ∈ V)
146brrelex2i 5331 . . . . . . . . 9 (𝐶* 𝐷𝐷 ∈ V)
15 unexg 7159 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵𝐷) ∈ V)
1613, 14, 15syl2an 589 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐵𝐷) ∈ V)
17163adant3 1162 . . . . . . 7 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐵𝐷) ∈ V)
1817adantr 472 . . . . . 6 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) → (𝐵𝐷) ∈ V)
19 elun 3917 . . . . . . . . . 10 (𝑦 ∈ (𝐴𝐶) ↔ (𝑦𝐴𝑦𝐶))
20 eqeq1 2769 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → (𝑎 = (𝑓𝑏) ↔ 𝑦 = (𝑓𝑏)))
2120rexbidv 3199 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (∃𝑏𝐵 𝑎 = (𝑓𝑏) ↔ ∃𝑏𝐵 𝑦 = (𝑓𝑏)))
2221rspcva 3460 . . . . . . . . . . . . . . 15 ((𝑦𝐴 ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → ∃𝑏𝐵 𝑦 = (𝑓𝑏))
23 fveq2 6377 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑧 → (𝑓𝑏) = (𝑓𝑧))
2423eqeq2d 2775 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑧 → (𝑦 = (𝑓𝑏) ↔ 𝑦 = (𝑓𝑧)))
2524cbvrexv 3320 . . . . . . . . . . . . . . . 16 (∃𝑏𝐵 𝑦 = (𝑓𝑏) ↔ ∃𝑧𝐵 𝑦 = (𝑓𝑧))
26 ssun1 3940 . . . . . . . . . . . . . . . . 17 𝐵 ⊆ (𝐵𝐷)
27 iftrue 4251 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝐵 → if(𝑧𝐵, 𝑓, 𝑔) = 𝑓)
2827fveq1d 6379 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝐵 → (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) = (𝑓𝑧))
2928eqeq2d 2775 . . . . . . . . . . . . . . . . . . 19 (𝑧𝐵 → (𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑓𝑧)))
3029biimprd 239 . . . . . . . . . . . . . . . . . 18 (𝑧𝐵 → (𝑦 = (𝑓𝑧) → 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
3130reximia 3155 . . . . . . . . . . . . . . . . 17 (∃𝑧𝐵 𝑦 = (𝑓𝑧) → ∃𝑧𝐵 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
32 ssrexv 3829 . . . . . . . . . . . . . . . . 17 (𝐵 ⊆ (𝐵𝐷) → (∃𝑧𝐵 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
3326, 31, 32mpsyl 68 . . . . . . . . . . . . . . . 16 (∃𝑧𝐵 𝑦 = (𝑓𝑧) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
3425, 33sylbi 208 . . . . . . . . . . . . . . 15 (∃𝑏𝐵 𝑦 = (𝑓𝑏) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
3522, 34syl 17 . . . . . . . . . . . . . 14 ((𝑦𝐴 ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
3635ancoms 450 . . . . . . . . . . . . 13 ((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ 𝑦𝐴) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
3736adantlr 706 . . . . . . . . . . . 12 (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦𝐴) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
3837adantll 705 . . . . . . . . . . 11 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦𝐴) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
39 eqeq1 2769 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → (𝑎 = (𝑔𝑏) ↔ 𝑦 = (𝑔𝑏)))
4039rexbidv 3199 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (∃𝑏𝐷 𝑎 = (𝑔𝑏) ↔ ∃𝑏𝐷 𝑦 = (𝑔𝑏)))
41 fveq2 6377 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑧 → (𝑔𝑏) = (𝑔𝑧))
4241eqeq2d 2775 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑧 → (𝑦 = (𝑔𝑏) ↔ 𝑦 = (𝑔𝑧)))
4342cbvrexv 3320 . . . . . . . . . . . . . . . 16 (∃𝑏𝐷 𝑦 = (𝑔𝑏) ↔ ∃𝑧𝐷 𝑦 = (𝑔𝑧))
4440, 43syl6bb 278 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → (∃𝑏𝐷 𝑎 = (𝑔𝑏) ↔ ∃𝑧𝐷 𝑦 = (𝑔𝑧)))
4544rspccva 3461 . . . . . . . . . . . . . 14 ((∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏) ∧ 𝑦𝐶) → ∃𝑧𝐷 𝑦 = (𝑔𝑧))
46 ssun2 3941 . . . . . . . . . . . . . . 15 𝐷 ⊆ (𝐵𝐷)
47 minel 4196 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧𝐷 ∧ (𝐵𝐷) = ∅) → ¬ 𝑧𝐵)
4847ancoms 450 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → ¬ 𝑧𝐵)
4948iffalsed 4256 . . . . . . . . . . . . . . . . . . . 20 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → if(𝑧𝐵, 𝑓, 𝑔) = 𝑔)
5049fveq1d 6379 . . . . . . . . . . . . . . . . . . 19 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) = (𝑔𝑧))
5150eqeq2d 2775 . . . . . . . . . . . . . . . . . 18 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → (𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑔𝑧)))
5251biimprd 239 . . . . . . . . . . . . . . . . 17 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → (𝑦 = (𝑔𝑧) → 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
5352reximdva 3163 . . . . . . . . . . . . . . . 16 ((𝐵𝐷) = ∅ → (∃𝑧𝐷 𝑦 = (𝑔𝑧) → ∃𝑧𝐷 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
5453imp 395 . . . . . . . . . . . . . . 15 (((𝐵𝐷) = ∅ ∧ ∃𝑧𝐷 𝑦 = (𝑔𝑧)) → ∃𝑧𝐷 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
55 ssrexv 3829 . . . . . . . . . . . . . . 15 (𝐷 ⊆ (𝐵𝐷) → (∃𝑧𝐷 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
5646, 54, 55mpsyl 68 . . . . . . . . . . . . . 14 (((𝐵𝐷) = ∅ ∧ ∃𝑧𝐷 𝑦 = (𝑔𝑧)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
5745, 56sylan2 586 . . . . . . . . . . . . 13 (((𝐵𝐷) = ∅ ∧ (∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏) ∧ 𝑦𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
5857anassrs 459 . . . . . . . . . . . 12 ((((𝐵𝐷) = ∅ ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦𝐶) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
5958adantlrl 711 . . . . . . . . . . 11 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦𝐶) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6038, 59jaodan 980 . . . . . . . . . 10 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ (𝑦𝐴𝑦𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6119, 60sylan2b 587 . . . . . . . . 9 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6261expl 449 . . . . . . . 8 ((𝐵𝐷) = ∅ → (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
63623ad2ant3 1165 . . . . . . 7 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
6463impl 447 . . . . . 6 ((((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6512, 18, 64wdom2d 8694 . . . . 5 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) → (𝐴𝐶) ≼* (𝐵𝐷))
6665expr 448 . . . 4 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → (∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏) → (𝐴𝐶) ≼* (𝐵𝐷)))
6766exlimdv 2028 . . 3 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → (∃𝑔𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏) → (𝐴𝐶) ≼* (𝐵𝐷)))
685, 67mpd 15 . 2 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → (𝐴𝐶) ≼* (𝐵𝐷))
692, 68exlimddv 2030 1 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≼* (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 873  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  cun 3732  cin 3733  wss 3734  c0 4081  ifcif 4245   class class class wbr 4811  cfv 6070  * cwdom 8671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-er 7949  df-en 8163  df-dom 8164  df-sdom 8165  df-wdom 8673
This theorem is referenced by: (None)
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