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Theorem brwdom3 9443
Description: Condition for weak dominance with a condition reminiscent of wdomd 9442. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
brwdom3 ((𝑋𝑉𝑌𝑊) → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
Distinct variable groups:   𝑓,𝑋,𝑥,𝑦   𝑓,𝑌,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑓)   𝑊(𝑥,𝑦,𝑓)

Proof of Theorem brwdom3
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3460 . 2 (𝑋𝑉𝑋 ∈ V)
2 elex 3460 . 2 (𝑌𝑊𝑌 ∈ V)
3 brwdom2 9434 . . . . 5 (𝑌 ∈ V → (𝑋* 𝑌 ↔ ∃𝑧 ∈ 𝒫 𝑌𝑓 𝑓:𝑧onto𝑋))
43adantl 483 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑧 ∈ 𝒫 𝑌𝑓 𝑓:𝑧onto𝑋))
5 dffo3 7038 . . . . . . . 8 (𝑓:𝑧onto𝑋 ↔ (𝑓:𝑧𝑋 ∧ ∀𝑥𝑋𝑦𝑧 𝑥 = (𝑓𝑦)))
65simprbi 498 . . . . . . 7 (𝑓:𝑧onto𝑋 → ∀𝑥𝑋𝑦𝑧 𝑥 = (𝑓𝑦))
7 elpwi 4558 . . . . . . . . . 10 (𝑧 ∈ 𝒫 𝑌𝑧𝑌)
8 ssrexv 4002 . . . . . . . . . 10 (𝑧𝑌 → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
97, 8syl 17 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝑌 → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
109adantl 483 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
1110ralimdv 3163 . . . . . . 7 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (∀𝑥𝑋𝑦𝑧 𝑥 = (𝑓𝑦) → ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
126, 11syl5 34 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (𝑓:𝑧onto𝑋 → ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
1312eximdv 1920 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (∃𝑓 𝑓:𝑧onto𝑋 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
1413rexlimdva 3149 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (∃𝑧 ∈ 𝒫 𝑌𝑓 𝑓:𝑧onto𝑋 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
154, 14sylbid 239 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
16 simpll 765 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → 𝑋 ∈ V)
17 simplr 767 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → 𝑌 ∈ V)
18 eqeq1 2741 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = (𝑓𝑦) ↔ 𝑧 = (𝑓𝑦)))
1918rexbidv 3172 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∃𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∃𝑦𝑌 𝑧 = (𝑓𝑦)))
20 fveq2 6829 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑓𝑦) = (𝑓𝑤))
2120eqeq2d 2748 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑧 = (𝑓𝑦) ↔ 𝑧 = (𝑓𝑤)))
2221cbvrexvw 3223 . . . . . . . . . . 11 (∃𝑦𝑌 𝑧 = (𝑓𝑦) ↔ ∃𝑤𝑌 𝑧 = (𝑓𝑤))
2319, 22bitrdi 287 . . . . . . . . . 10 (𝑥 = 𝑧 → (∃𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∃𝑤𝑌 𝑧 = (𝑓𝑤)))
2423cbvralvw 3222 . . . . . . . . 9 (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2524biimpi 215 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) → ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2625adantl 483 . . . . . . 7 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2726r19.21bi 3231 . . . . . 6 ((((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) ∧ 𝑧𝑋) → ∃𝑤𝑌 𝑧 = (𝑓𝑤))
2816, 17, 27wdom2d 9441 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → 𝑋* 𝑌)
2928ex 414 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) → 𝑋* 𝑌))
3029exlimdv 1936 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) → 𝑋* 𝑌))
3115, 30impbid 211 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
321, 2, 31syl2an 597 1 ((𝑋𝑉𝑌𝑊) → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1541  wex 1781  wcel 2106  wral 3062  wrex 3071  Vcvv 3442  wss 3901  𝒫 cpw 4551   class class class wbr 5096  wf 6479  ontowfo 6481  cfv 6483  * cwdom 9425
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 2708  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-br 5097  df-opab 5159  df-mpt 5180  df-id 5522  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-en 8809  df-dom 8810  df-sdom 8811  df-wdom 9426
This theorem is referenced by:  brwdom3i  9444
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