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Theorem unwdomg 8434
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 8433 . . 3 (𝐴* 𝐵 → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
213ad2ant1 1080 . 2 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
3 brwdom3i 8433 . . . . 5 (𝐶* 𝐷 → ∃𝑔𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))
433ad2ant2 1081 . . . 4 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → ∃𝑔𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))
54adantr 481 . . 3 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → ∃𝑔𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))
6 relwdom 8416 . . . . . . . . . 10 Rel ≼*
76brrelexi 5123 . . . . . . . . 9 (𝐴* 𝐵𝐴 ∈ V)
86brrelexi 5123 . . . . . . . . 9 (𝐶* 𝐷𝐶 ∈ V)
9 unexg 6913 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴𝐶) ∈ V)
107, 8, 9syl2an 494 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐴𝐶) ∈ V)
11103adant3 1079 . . . . . . 7 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ∈ V)
1211adantr 481 . . . . . 6 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) → (𝐴𝐶) ∈ V)
136brrelex2i 5124 . . . . . . . . 9 (𝐴* 𝐵𝐵 ∈ V)
146brrelex2i 5124 . . . . . . . . 9 (𝐶* 𝐷𝐷 ∈ V)
15 unexg 6913 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵𝐷) ∈ V)
1613, 14, 15syl2an 494 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐵𝐷) ∈ V)
17163adant3 1079 . . . . . . 7 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐵𝐷) ∈ V)
1817adantr 481 . . . . . 6 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) → (𝐵𝐷) ∈ V)
19 elun 3736 . . . . . . . . . 10 (𝑦 ∈ (𝐴𝐶) ↔ (𝑦𝐴𝑦𝐶))
20 eqeq1 2630 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → (𝑎 = (𝑓𝑏) ↔ 𝑦 = (𝑓𝑏)))
2120rexbidv 3050 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (∃𝑏𝐵 𝑎 = (𝑓𝑏) ↔ ∃𝑏𝐵 𝑦 = (𝑓𝑏)))
2221rspcva 3298 . . . . . . . . . . . . . . 15 ((𝑦𝐴 ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → ∃𝑏𝐵 𝑦 = (𝑓𝑏))
23 fveq2 6150 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑧 → (𝑓𝑏) = (𝑓𝑧))
2423eqeq2d 2636 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑧 → (𝑦 = (𝑓𝑏) ↔ 𝑦 = (𝑓𝑧)))
2524cbvrexv 3165 . . . . . . . . . . . . . . . 16 (∃𝑏𝐵 𝑦 = (𝑓𝑏) ↔ ∃𝑧𝐵 𝑦 = (𝑓𝑧))
26 ssun1 3759 . . . . . . . . . . . . . . . . 17 𝐵 ⊆ (𝐵𝐷)
27 iftrue 4069 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝐵 → if(𝑧𝐵, 𝑓, 𝑔) = 𝑓)
2827fveq1d 6152 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝐵 → (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) = (𝑓𝑧))
2928eqeq2d 2636 . . . . . . . . . . . . . . . . . . 19 (𝑧𝐵 → (𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑓𝑧)))
3029biimprd 238 . . . . . . . . . . . . . . . . . 18 (𝑧𝐵 → (𝑦 = (𝑓𝑧) → 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
3130reximia 3008 . . . . . . . . . . . . . . . . 17 (∃𝑧𝐵 𝑦 = (𝑓𝑧) → ∃𝑧𝐵 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
32 ssrexv 3651 . . . . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . 13 ((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ 𝑦𝐴) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
3736adantlr 750 . . . . . . . . . . . 12 (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦𝐴) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
3837adantll 749 . . . . . . . . . . 11 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦𝐴) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
39 eqeq1 2630 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → (𝑎 = (𝑔𝑏) ↔ 𝑦 = (𝑔𝑏)))
4039rexbidv 3050 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (∃𝑏𝐷 𝑎 = (𝑔𝑏) ↔ ∃𝑏𝐷 𝑦 = (𝑔𝑏)))
41 fveq2 6150 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑧 → (𝑔𝑏) = (𝑔𝑧))
4241eqeq2d 2636 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑧 → (𝑦 = (𝑔𝑏) ↔ 𝑦 = (𝑔𝑧)))
4342cbvrexv 3165 . . . . . . . . . . . . . . . 16 (∃𝑏𝐷 𝑦 = (𝑔𝑏) ↔ ∃𝑧𝐷 𝑦 = (𝑔𝑧))
4440, 43syl6bb 276 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → (∃𝑏𝐷 𝑎 = (𝑔𝑏) ↔ ∃𝑧𝐷 𝑦 = (𝑔𝑧)))
4544rspccva 3299 . . . . . . . . . . . . . 14 ((∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏) ∧ 𝑦𝐶) → ∃𝑧𝐷 𝑦 = (𝑔𝑧))
46 ssun2 3760 . . . . . . . . . . . . . . 15 𝐷 ⊆ (𝐵𝐷)
47 minel 4010 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧𝐷 ∧ (𝐵𝐷) = ∅) → ¬ 𝑧𝐵)
4847ancoms 469 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → ¬ 𝑧𝐵)
4948iffalsed 4074 . . . . . . . . . . . . . . . . . . . 20 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → if(𝑧𝐵, 𝑓, 𝑔) = 𝑔)
5049fveq1d 6152 . . . . . . . . . . . . . . . . . . 19 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) = (𝑔𝑧))
5150eqeq2d 2636 . . . . . . . . . . . . . . . . . 18 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → (𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑔𝑧)))
5251biimprd 238 . . . . . . . . . . . . . . . . 17 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → (𝑦 = (𝑔𝑧) → 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
5352reximdva 3016 . . . . . . . . . . . . . . . 16 ((𝐵𝐷) = ∅ → (∃𝑧𝐷 𝑦 = (𝑔𝑧) → ∃𝑧𝐷 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
5453imp 445 . . . . . . . . . . . . . . 15 (((𝐵𝐷) = ∅ ∧ ∃𝑧𝐷 𝑦 = (𝑔𝑧)) → ∃𝑧𝐷 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
55 ssrexv 3651 . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . 13 (((𝐵𝐷) = ∅ ∧ (∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏) ∧ 𝑦𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
5857anassrs 679 . . . . . . . . . . . 12 ((((𝐵𝐷) = ∅ ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦𝐶) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
5958adantlrl 755 . . . . . . . . . . 11 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦𝐶) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6038, 59jaodan 825 . . . . . . . . . 10 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ (𝑦𝐴𝑦𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6119, 60sylan2b 492 . . . . . . . . 9 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6261expl 647 . . . . . . . 8 ((𝐵𝐷) = ∅ → (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
63623ad2ant3 1082 . . . . . . 7 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
6463impl 649 . . . . . 6 ((((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6512, 18, 64wdom2d 8430 . . . . 5 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) → (𝐴𝐶) ≼* (𝐵𝐷))
6665expr 642 . . . 4 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → (∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏) → (𝐴𝐶) ≼* (𝐵𝐷)))
6766exlimdv 1863 . . 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 1865 1 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≼* (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1992  wral 2912  wrex 2913  Vcvv 3191  cun 3558  cin 3559  wss 3560  c0 3896  ifcif 4063   class class class wbr 4618  cfv 5850  * cwdom 8407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-wdom 8409
This theorem is referenced by: (None)
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