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Theorem brwdom3 9518
Description: Condition for weak dominance with a condition reminiscent of wdomd 9517. (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 3463 . 2 (𝑋𝑉𝑋 ∈ V)
2 elex 3463 . 2 (𝑌𝑊𝑌 ∈ V)
3 brwdom2 9509 . . . . 5 (𝑌 ∈ V → (𝑋* 𝑌 ↔ ∃𝑧 ∈ 𝒫 𝑌𝑓 𝑓:𝑧onto𝑋))
43adantl 482 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑧 ∈ 𝒫 𝑌𝑓 𝑓:𝑧onto𝑋))
5 dffo3 7052 . . . . . . . 8 (𝑓:𝑧onto𝑋 ↔ (𝑓:𝑧𝑋 ∧ ∀𝑥𝑋𝑦𝑧 𝑥 = (𝑓𝑦)))
65simprbi 497 . . . . . . 7 (𝑓:𝑧onto𝑋 → ∀𝑥𝑋𝑦𝑧 𝑥 = (𝑓𝑦))
7 elpwi 4567 . . . . . . . . . 10 (𝑧 ∈ 𝒫 𝑌𝑧𝑌)
8 ssrexv 4011 . . . . . . . . . 10 (𝑧𝑌 → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
97, 8syl 17 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝑌 → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
109adantl 482 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
1110ralimdv 3166 . . . . . . 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 3152 . . . 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 2740 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = (𝑓𝑦) ↔ 𝑧 = (𝑓𝑦)))
1918rexbidv 3175 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∃𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∃𝑦𝑌 𝑧 = (𝑓𝑦)))
20 fveq2 6842 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑓𝑦) = (𝑓𝑤))
2120eqeq2d 2747 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑧 = (𝑓𝑦) ↔ 𝑧 = (𝑓𝑤)))
2221cbvrexvw 3226 . . . . . . . . . . 11 (∃𝑦𝑌 𝑧 = (𝑓𝑦) ↔ ∃𝑤𝑌 𝑧 = (𝑓𝑤))
2319, 22bitrdi 286 . . . . . . . . . 10 (𝑥 = 𝑧 → (∃𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∃𝑤𝑌 𝑧 = (𝑓𝑤)))
2423cbvralvw 3225 . . . . . . . . 9 (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2524biimpi 215 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) → ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2625adantl 482 . . . . . . 7 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2726r19.21bi 3234 . . . . . 6 ((((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) ∧ 𝑧𝑋) → ∃𝑤𝑌 𝑧 = (𝑓𝑤))
2816, 17, 27wdom2d 9516 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → 𝑋* 𝑌)
2928ex 413 . . . 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 596 1 ((𝑋𝑉𝑌𝑊) → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910  𝒫 cpw 4560   class class class wbr 5105  wf 6492  ontowfo 6494  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:  brwdom3i  9519
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