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Theorem brwdom3 9579
Description: Condition for weak dominance with a condition reminiscent of wdomd 9578. (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 3491 . 2 (𝑋𝑉𝑋 ∈ V)
2 elex 3491 . 2 (𝑌𝑊𝑌 ∈ V)
3 brwdom2 9570 . . . . 5 (𝑌 ∈ V → (𝑋* 𝑌 ↔ ∃𝑧 ∈ 𝒫 𝑌𝑓 𝑓:𝑧onto𝑋))
43adantl 480 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑧 ∈ 𝒫 𝑌𝑓 𝑓:𝑧onto𝑋))
5 dffo3 7102 . . . . . . . 8 (𝑓:𝑧onto𝑋 ↔ (𝑓:𝑧𝑋 ∧ ∀𝑥𝑋𝑦𝑧 𝑥 = (𝑓𝑦)))
65simprbi 495 . . . . . . 7 (𝑓:𝑧onto𝑋 → ∀𝑥𝑋𝑦𝑧 𝑥 = (𝑓𝑦))
7 elpwi 4608 . . . . . . . . . 10 (𝑧 ∈ 𝒫 𝑌𝑧𝑌)
8 ssrexv 4050 . . . . . . . . . 10 (𝑧𝑌 → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
97, 8syl 17 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝑌 → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
109adantl 480 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
1110ralimdv 3167 . . . . . . 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 1918 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (∃𝑓 𝑓:𝑧onto𝑋 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
1413rexlimdva 3153 . . . 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 763 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → 𝑋 ∈ V)
17 simplr 765 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → 𝑌 ∈ V)
18 eqeq1 2734 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = (𝑓𝑦) ↔ 𝑧 = (𝑓𝑦)))
1918rexbidv 3176 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∃𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∃𝑦𝑌 𝑧 = (𝑓𝑦)))
20 fveq2 6890 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑓𝑦) = (𝑓𝑤))
2120eqeq2d 2741 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑧 = (𝑓𝑦) ↔ 𝑧 = (𝑓𝑤)))
2221cbvrexvw 3233 . . . . . . . . . . 11 (∃𝑦𝑌 𝑧 = (𝑓𝑦) ↔ ∃𝑤𝑌 𝑧 = (𝑓𝑤))
2319, 22bitrdi 286 . . . . . . . . . 10 (𝑥 = 𝑧 → (∃𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∃𝑤𝑌 𝑧 = (𝑓𝑤)))
2423cbvralvw 3232 . . . . . . . . 9 (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2524biimpi 215 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) → ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2625adantl 480 . . . . . . 7 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2726r19.21bi 3246 . . . . . 6 ((((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) ∧ 𝑧𝑋) → ∃𝑤𝑌 𝑧 = (𝑓𝑤))
2816, 17, 27wdom2d 9577 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → 𝑋* 𝑌)
2928ex 411 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) → 𝑋* 𝑌))
3029exlimdv 1934 . . 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 594 1 ((𝑋𝑉𝑌𝑊) → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wex 1779  wcel 2104  wral 3059  wrex 3068  Vcvv 3472  wss 3947  𝒫 cpw 4601   class class class wbr 5147  wf 6538  ontowfo 6540  cfv 6542  * cwdom 9561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-en 8942  df-dom 8943  df-sdom 8944  df-wdom 9562
This theorem is referenced by:  brwdom3i  9580
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