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Theorem unwdomg 9520
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 9519 . . 3 (𝐴* 𝐵 → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
213ad2ant1 1133 . 2 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
3 brwdom3i 9519 . . . . 5 (𝐶* 𝐷 → ∃𝑔𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))
433ad2ant2 1134 . . . 4 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → ∃𝑔𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))
54adantr 481 . . 3 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → ∃𝑔𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))
6 relwdom 9502 . . . . . . . . . 10 Rel ≼*
76brrelex1i 5688 . . . . . . . . 9 (𝐴* 𝐵𝐴 ∈ V)
86brrelex1i 5688 . . . . . . . . 9 (𝐶* 𝐷𝐶 ∈ V)
9 unexg 7683 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴𝐶) ∈ V)
107, 8, 9syl2an 596 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐴𝐶) ∈ V)
11103adant3 1132 . . . . . . 7 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ∈ V)
1211adantr 481 . . . . . 6 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) → (𝐴𝐶) ∈ V)
136brrelex2i 5689 . . . . . . . . 9 (𝐴* 𝐵𝐵 ∈ V)
146brrelex2i 5689 . . . . . . . . 9 (𝐶* 𝐷𝐷 ∈ V)
15 unexg 7683 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵𝐷) ∈ V)
1613, 14, 15syl2an 596 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐵𝐷) ∈ V)
17163adant3 1132 . . . . . . 7 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐵𝐷) ∈ V)
1817adantr 481 . . . . . 6 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) → (𝐵𝐷) ∈ V)
19 elun 4108 . . . . . . . . . 10 (𝑦 ∈ (𝐴𝐶) ↔ (𝑦𝐴𝑦𝐶))
20 eqeq1 2740 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → (𝑎 = (𝑓𝑏) ↔ 𝑦 = (𝑓𝑏)))
2120rexbidv 3175 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (∃𝑏𝐵 𝑎 = (𝑓𝑏) ↔ ∃𝑏𝐵 𝑦 = (𝑓𝑏)))
2221rspcva 3579 . . . . . . . . . . . . . . 15 ((𝑦𝐴 ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → ∃𝑏𝐵 𝑦 = (𝑓𝑏))
23 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑧 → (𝑓𝑏) = (𝑓𝑧))
2423eqeq2d 2747 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑧 → (𝑦 = (𝑓𝑏) ↔ 𝑦 = (𝑓𝑧)))
2524cbvrexvw 3226 . . . . . . . . . . . . . . . 16 (∃𝑏𝐵 𝑦 = (𝑓𝑏) ↔ ∃𝑧𝐵 𝑦 = (𝑓𝑧))
26 ssun1 4132 . . . . . . . . . . . . . . . . 17 𝐵 ⊆ (𝐵𝐷)
27 iftrue 4492 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝐵 → if(𝑧𝐵, 𝑓, 𝑔) = 𝑓)
2827fveq1d 6844 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝐵 → (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) = (𝑓𝑧))
2928eqeq2d 2747 . . . . . . . . . . . . . . . . . . 19 (𝑧𝐵 → (𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑓𝑧)))
3029biimprd 247 . . . . . . . . . . . . . . . . . 18 (𝑧𝐵 → (𝑦 = (𝑓𝑧) → 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
3130reximia 3084 . . . . . . . . . . . . . . . . 17 (∃𝑧𝐵 𝑦 = (𝑓𝑧) → ∃𝑧𝐵 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
32 ssrexv 4011 . . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . 13 ((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ 𝑦𝐴) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
3736adantlr 713 . . . . . . . . . . . 12 (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦𝐴) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
3837adantll 712 . . . . . . . . . . 11 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦𝐴) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
39 eqeq1 2740 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → (𝑎 = (𝑔𝑏) ↔ 𝑦 = (𝑔𝑏)))
4039rexbidv 3175 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (∃𝑏𝐷 𝑎 = (𝑔𝑏) ↔ ∃𝑏𝐷 𝑦 = (𝑔𝑏)))
41 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑧 → (𝑔𝑏) = (𝑔𝑧))
4241eqeq2d 2747 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑧 → (𝑦 = (𝑔𝑏) ↔ 𝑦 = (𝑔𝑧)))
4342cbvrexvw 3226 . . . . . . . . . . . . . . . 16 (∃𝑏𝐷 𝑦 = (𝑔𝑏) ↔ ∃𝑧𝐷 𝑦 = (𝑔𝑧))
4440, 43bitrdi 286 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → (∃𝑏𝐷 𝑎 = (𝑔𝑏) ↔ ∃𝑧𝐷 𝑦 = (𝑔𝑧)))
4544rspccva 3580 . . . . . . . . . . . . . 14 ((∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏) ∧ 𝑦𝐶) → ∃𝑧𝐷 𝑦 = (𝑔𝑧))
46 ssun2 4133 . . . . . . . . . . . . . . 15 𝐷 ⊆ (𝐵𝐷)
47 minel 4425 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧𝐷 ∧ (𝐵𝐷) = ∅) → ¬ 𝑧𝐵)
4847ancoms 459 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → ¬ 𝑧𝐵)
4948iffalsed 4497 . . . . . . . . . . . . . . . . . . . 20 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → if(𝑧𝐵, 𝑓, 𝑔) = 𝑔)
5049fveq1d 6844 . . . . . . . . . . . . . . . . . . 19 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) = (𝑔𝑧))
5150eqeq2d 2747 . . . . . . . . . . . . . . . . . 18 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → (𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑔𝑧)))
5251biimprd 247 . . . . . . . . . . . . . . . . 17 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → (𝑦 = (𝑔𝑧) → 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
5352reximdva 3165 . . . . . . . . . . . . . . . 16 ((𝐵𝐷) = ∅ → (∃𝑧𝐷 𝑦 = (𝑔𝑧) → ∃𝑧𝐷 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
5453imp 407 . . . . . . . . . . . . . . 15 (((𝐵𝐷) = ∅ ∧ ∃𝑧𝐷 𝑦 = (𝑔𝑧)) → ∃𝑧𝐷 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
55 ssrexv 4011 . . . . . . . . . . . . . . 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 593 . . . . . . . . . . . . 13 (((𝐵𝐷) = ∅ ∧ (∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏) ∧ 𝑦𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
5857anassrs 468 . . . . . . . . . . . 12 ((((𝐵𝐷) = ∅ ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦𝐶) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
5958adantlrl 718 . . . . . . . . . . 11 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦𝐶) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6038, 59jaodan 956 . . . . . . . . . 10 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ (𝑦𝐴𝑦𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6119, 60sylan2b 594 . . . . . . . . 9 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6261expl 458 . . . . . . . 8 ((𝐵𝐷) = ∅ → (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
63623ad2ant3 1135 . . . . . . 7 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
6463impl 456 . . . . . 6 ((((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6512, 18, 64wdom2d 9516 . . . . 5 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) → (𝐴𝐶) ≼* (𝐵𝐷))
6665expr 457 . . . 4 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → (∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏) → (𝐴𝐶) ≼* (𝐵𝐷)))
6766exlimdv 1936 . . 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 1938 1 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≼* (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  cun 3908  cin 3909  wss 3910  c0 4282  ifcif 4486   class class class wbr 5105  cfv 6496  * cwdom 9500
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-en 8884  df-dom 8885  df-sdom 8886  df-wdom 9501
This theorem is referenced by: (None)
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