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Theorem brwdom3 9034
Description: Condition for weak dominance with a condition reminiscent of wdomd 9033. (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 3487 . 2 (𝑋𝑉𝑋 ∈ V)
2 elex 3487 . 2 (𝑌𝑊𝑌 ∈ V)
3 brwdom2 9025 . . . . 5 (𝑌 ∈ V → (𝑋* 𝑌 ↔ ∃𝑧 ∈ 𝒫 𝑌𝑓 𝑓:𝑧onto𝑋))
43adantl 485 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑧 ∈ 𝒫 𝑌𝑓 𝑓:𝑧onto𝑋))
5 dffo3 6850 . . . . . . . 8 (𝑓:𝑧onto𝑋 ↔ (𝑓:𝑧𝑋 ∧ ∀𝑥𝑋𝑦𝑧 𝑥 = (𝑓𝑦)))
65simprbi 500 . . . . . . 7 (𝑓:𝑧onto𝑋 → ∀𝑥𝑋𝑦𝑧 𝑥 = (𝑓𝑦))
7 elpwi 4520 . . . . . . . . . 10 (𝑧 ∈ 𝒫 𝑌𝑧𝑌)
8 ssrexv 4009 . . . . . . . . . 10 (𝑧𝑌 → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
97, 8syl 17 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝑌 → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
109adantl 485 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
1110ralimdv 3170 . . . . . . 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 3270 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (∃𝑧 ∈ 𝒫 𝑌𝑓 𝑓:𝑧onto𝑋 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
154, 14sylbid 243 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
16 simpll 766 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → 𝑋 ∈ V)
17 simplr 768 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → 𝑌 ∈ V)
18 eqeq1 2826 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = (𝑓𝑦) ↔ 𝑧 = (𝑓𝑦)))
1918rexbidv 3283 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∃𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∃𝑦𝑌 𝑧 = (𝑓𝑦)))
20 fveq2 6652 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑓𝑦) = (𝑓𝑤))
2120eqeq2d 2833 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑧 = (𝑓𝑦) ↔ 𝑧 = (𝑓𝑤)))
2221cbvrexvw 3425 . . . . . . . . . . 11 (∃𝑦𝑌 𝑧 = (𝑓𝑦) ↔ ∃𝑤𝑌 𝑧 = (𝑓𝑤))
2319, 22syl6bb 290 . . . . . . . . . 10 (𝑥 = 𝑧 → (∃𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∃𝑤𝑌 𝑧 = (𝑓𝑤)))
2423cbvralvw 3424 . . . . . . . . 9 (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2524biimpi 219 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) → ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2625adantl 485 . . . . . . 7 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2726r19.21bi 3198 . . . . . 6 ((((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) ∧ 𝑧𝑋) → ∃𝑤𝑌 𝑧 = (𝑓𝑤))
2816, 17, 27wdom2d 9032 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → 𝑋* 𝑌)
2928ex 416 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) → 𝑋* 𝑌))
3029exlimdv 1934 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) → 𝑋* 𝑌))
3115, 30impbid 215 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
321, 2, 31syl2an 598 1 ((𝑋𝑉𝑌𝑊) → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2114  wral 3130  wrex 3131  Vcvv 3469  wss 3908  𝒫 cpw 4511   class class class wbr 5042  wf 6330  ontowfo 6332  cfv 6334  * cwdom 9016
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-wdom 9017
This theorem is referenced by:  brwdom3i  9035
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