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Theorem brwdom3 8841
Description: Condition for weak dominance with a condition reminiscent of wdomd 8840. (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 3434 . 2 (𝑋𝑉𝑋 ∈ V)
2 elex 3434 . 2 (𝑌𝑊𝑌 ∈ V)
3 brwdom2 8832 . . . . 5 (𝑌 ∈ V → (𝑋* 𝑌 ↔ ∃𝑧 ∈ 𝒫 𝑌𝑓 𝑓:𝑧onto𝑋))
43adantl 474 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑧 ∈ 𝒫 𝑌𝑓 𝑓:𝑧onto𝑋))
5 dffo3 6691 . . . . . . . 8 (𝑓:𝑧onto𝑋 ↔ (𝑓:𝑧𝑋 ∧ ∀𝑥𝑋𝑦𝑧 𝑥 = (𝑓𝑦)))
65simprbi 489 . . . . . . 7 (𝑓:𝑧onto𝑋 → ∀𝑥𝑋𝑦𝑧 𝑥 = (𝑓𝑦))
7 elpwi 4432 . . . . . . . . . 10 (𝑧 ∈ 𝒫 𝑌𝑧𝑌)
8 ssrexv 3925 . . . . . . . . . 10 (𝑧𝑌 → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
97, 8syl 17 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝑌 → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
109adantl 474 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (∃𝑦𝑧 𝑥 = (𝑓𝑦) → ∃𝑦𝑌 𝑥 = (𝑓𝑦)))
1110ralimdv 3129 . . . . . . 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 1876 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑧 ∈ 𝒫 𝑌) → (∃𝑓 𝑓:𝑧onto𝑋 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
1413rexlimdva 3230 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (∃𝑧 ∈ 𝒫 𝑌𝑓 𝑓:𝑧onto𝑋 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
154, 14sylbid 232 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
16 simpll 754 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → 𝑋 ∈ V)
17 simplr 756 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → 𝑌 ∈ V)
18 eqeq1 2783 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = (𝑓𝑦) ↔ 𝑧 = (𝑓𝑦)))
1918rexbidv 3243 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∃𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∃𝑦𝑌 𝑧 = (𝑓𝑦)))
20 fveq2 6499 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑓𝑦) = (𝑓𝑤))
2120eqeq2d 2789 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑧 = (𝑓𝑦) ↔ 𝑧 = (𝑓𝑤)))
2221cbvrexv 3385 . . . . . . . . . . 11 (∃𝑦𝑌 𝑧 = (𝑓𝑦) ↔ ∃𝑤𝑌 𝑧 = (𝑓𝑤))
2319, 22syl6bb 279 . . . . . . . . . 10 (𝑥 = 𝑧 → (∃𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∃𝑤𝑌 𝑧 = (𝑓𝑤)))
2423cbvralv 3384 . . . . . . . . 9 (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) ↔ ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2524biimpi 208 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) → ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2625adantl 474 . . . . . . 7 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → ∀𝑧𝑋𝑤𝑌 𝑧 = (𝑓𝑤))
2726r19.21bi 3159 . . . . . 6 ((((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) ∧ 𝑧𝑋) → ∃𝑤𝑌 𝑧 = (𝑓𝑤))
2816, 17, 27wdom2d 8839 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ ∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)) → 𝑋* 𝑌)
2928ex 405 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (∀𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) → 𝑋* 𝑌))
3029exlimdv 1892 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦) → 𝑋* 𝑌))
3115, 30impbid 204 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
321, 2, 31syl2an 586 1 ((𝑋𝑉𝑌𝑊) → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wex 1742  wcel 2050  wral 3089  wrex 3090  Vcvv 3416  wss 3830  𝒫 cpw 4422   class class class wbr 4929  wf 6184  ontowfo 6186  cfv 6188  * cwdom 8816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-er 8089  df-en 8307  df-dom 8308  df-sdom 8309  df-wdom 8818
This theorem is referenced by:  brwdom3i  8842
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