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Theorem unwdomg 9528
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 9527 . . 3 (𝐴* 𝐵 → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
213ad2ant1 1134 . 2 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → ∃𝑓𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏))
3 brwdom3i 9527 . . . . 5 (𝐶* 𝐷 → ∃𝑔𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))
433ad2ant2 1135 . . . 4 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → ∃𝑔𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))
54adantr 482 . . 3 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → ∃𝑔𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))
6 relwdom 9510 . . . . . . . . . 10 Rel ≼*
76brrelex1i 5692 . . . . . . . . 9 (𝐴* 𝐵𝐴 ∈ V)
86brrelex1i 5692 . . . . . . . . 9 (𝐶* 𝐷𝐶 ∈ V)
9 unexg 7687 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴𝐶) ∈ V)
107, 8, 9syl2an 597 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐴𝐶) ∈ V)
11103adant3 1133 . . . . . . 7 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ∈ V)
1211adantr 482 . . . . . 6 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) → (𝐴𝐶) ∈ V)
136brrelex2i 5693 . . . . . . . . 9 (𝐴* 𝐵𝐵 ∈ V)
146brrelex2i 5693 . . . . . . . . 9 (𝐶* 𝐷𝐷 ∈ V)
15 unexg 7687 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵𝐷) ∈ V)
1613, 14, 15syl2an 597 . . . . . . . 8 ((𝐴* 𝐵𝐶* 𝐷) → (𝐵𝐷) ∈ V)
17163adant3 1133 . . . . . . 7 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐵𝐷) ∈ V)
1817adantr 482 . . . . . 6 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) → (𝐵𝐷) ∈ V)
19 elun 4112 . . . . . . . . . 10 (𝑦 ∈ (𝐴𝐶) ↔ (𝑦𝐴𝑦𝐶))
20 eqeq1 2737 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → (𝑎 = (𝑓𝑏) ↔ 𝑦 = (𝑓𝑏)))
2120rexbidv 3172 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (∃𝑏𝐵 𝑎 = (𝑓𝑏) ↔ ∃𝑏𝐵 𝑦 = (𝑓𝑏)))
2221rspcva 3581 . . . . . . . . . . . . . . 15 ((𝑦𝐴 ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → ∃𝑏𝐵 𝑦 = (𝑓𝑏))
23 fveq2 6846 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑧 → (𝑓𝑏) = (𝑓𝑧))
2423eqeq2d 2744 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑧 → (𝑦 = (𝑓𝑏) ↔ 𝑦 = (𝑓𝑧)))
2524cbvrexvw 3225 . . . . . . . . . . . . . . . 16 (∃𝑏𝐵 𝑦 = (𝑓𝑏) ↔ ∃𝑧𝐵 𝑦 = (𝑓𝑧))
26 ssun1 4136 . . . . . . . . . . . . . . . . 17 𝐵 ⊆ (𝐵𝐷)
27 iftrue 4496 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝐵 → if(𝑧𝐵, 𝑓, 𝑔) = 𝑓)
2827fveq1d 6848 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝐵 → (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) = (𝑓𝑧))
2928eqeq2d 2744 . . . . . . . . . . . . . . . . . . 19 (𝑧𝐵 → (𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑓𝑧)))
3029biimprd 248 . . . . . . . . . . . . . . . . . 18 (𝑧𝐵 → (𝑦 = (𝑓𝑧) → 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
3130reximia 3081 . . . . . . . . . . . . . . . . 17 (∃𝑧𝐵 𝑦 = (𝑓𝑧) → ∃𝑧𝐵 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
32 ssrexv 4015 . . . . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . 13 ((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ 𝑦𝐴) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
3736adantlr 714 . . . . . . . . . . . 12 (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦𝐴) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
3837adantll 713 . . . . . . . . . . 11 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦𝐴) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
39 eqeq1 2737 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑦 → (𝑎 = (𝑔𝑏) ↔ 𝑦 = (𝑔𝑏)))
4039rexbidv 3172 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (∃𝑏𝐷 𝑎 = (𝑔𝑏) ↔ ∃𝑏𝐷 𝑦 = (𝑔𝑏)))
41 fveq2 6846 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑧 → (𝑔𝑏) = (𝑔𝑧))
4241eqeq2d 2744 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑧 → (𝑦 = (𝑔𝑏) ↔ 𝑦 = (𝑔𝑧)))
4342cbvrexvw 3225 . . . . . . . . . . . . . . . 16 (∃𝑏𝐷 𝑦 = (𝑔𝑏) ↔ ∃𝑧𝐷 𝑦 = (𝑔𝑧))
4440, 43bitrdi 287 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → (∃𝑏𝐷 𝑎 = (𝑔𝑏) ↔ ∃𝑧𝐷 𝑦 = (𝑔𝑧)))
4544rspccva 3582 . . . . . . . . . . . . . 14 ((∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏) ∧ 𝑦𝐶) → ∃𝑧𝐷 𝑦 = (𝑔𝑧))
46 ssun2 4137 . . . . . . . . . . . . . . 15 𝐷 ⊆ (𝐵𝐷)
47 minel 4429 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧𝐷 ∧ (𝐵𝐷) = ∅) → ¬ 𝑧𝐵)
4847ancoms 460 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → ¬ 𝑧𝐵)
4948iffalsed 4501 . . . . . . . . . . . . . . . . . . . 20 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → if(𝑧𝐵, 𝑓, 𝑔) = 𝑔)
5049fveq1d 6848 . . . . . . . . . . . . . . . . . . 19 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) = (𝑔𝑧))
5150eqeq2d 2744 . . . . . . . . . . . . . . . . . 18 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → (𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧) ↔ 𝑦 = (𝑔𝑧)))
5251biimprd 248 . . . . . . . . . . . . . . . . 17 (((𝐵𝐷) = ∅ ∧ 𝑧𝐷) → (𝑦 = (𝑔𝑧) → 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
5352reximdva 3162 . . . . . . . . . . . . . . . 16 ((𝐵𝐷) = ∅ → (∃𝑧𝐷 𝑦 = (𝑔𝑧) → ∃𝑧𝐷 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
5453imp 408 . . . . . . . . . . . . . . 15 (((𝐵𝐷) = ∅ ∧ ∃𝑧𝐷 𝑦 = (𝑔𝑧)) → ∃𝑧𝐷 𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
55 ssrexv 4015 . . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . 13 (((𝐵𝐷) = ∅ ∧ (∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏) ∧ 𝑦𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
5857anassrs 469 . . . . . . . . . . . 12 ((((𝐵𝐷) = ∅ ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦𝐶) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
5958adantlrl 719 . . . . . . . . . . 11 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦𝐶) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6038, 59jaodan 957 . . . . . . . . . 10 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ (𝑦𝐴𝑦𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6119, 60sylan2b 595 . . . . . . . . 9 ((((𝐵𝐷) = ∅ ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6261expl 459 . . . . . . . 8 ((𝐵𝐷) = ∅ → (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
63623ad2ant3 1136 . . . . . . 7 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (((∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏)) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧)))
6463impl 457 . . . . . 6 ((((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) ∧ 𝑦 ∈ (𝐴𝐶)) → ∃𝑧 ∈ (𝐵𝐷)𝑦 = (if(𝑧𝐵, 𝑓, 𝑔)‘𝑧))
6512, 18, 64wdom2d 9524 . . . . 5 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ (∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏) ∧ ∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏))) → (𝐴𝐶) ≼* (𝐵𝐷))
6665expr 458 . . . 4 (((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 = (𝑓𝑏)) → (∀𝑎𝐶𝑏𝐷 𝑎 = (𝑔𝑏) → (𝐴𝐶) ≼* (𝐵𝐷)))
6766exlimdv 1937 . . 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 1939 1 ((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≼* (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846  w3a 1088   = wceq 1542  wex 1782  wcel 2107  wral 3061  wrex 3070  Vcvv 3447  cun 3912  cin 3913  wss 3914  c0 4286  ifcif 4490   class class class wbr 5109  cfv 6500  * cwdom 9508
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-en 8890  df-dom 8891  df-sdom 8892  df-wdom 9509
This theorem is referenced by: (None)
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