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Theorem wdompwdom 8434
Description: Weak dominance strengthens to usual dominance on the power sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
wdompwdom (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)

Proof of Theorem wdompwdom
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 relwdom 8422 . . . . . 6 Rel ≼*
21brrelex2i 5124 . . . . 5 (𝑋* 𝑌𝑌 ∈ V)
3 pwexg 4815 . . . . 5 (𝑌 ∈ V → 𝒫 𝑌 ∈ V)
42, 3syl 17 . . . 4 (𝑋* 𝑌 → 𝒫 𝑌 ∈ V)
5 0ss 3949 . . . . 5 ∅ ⊆ 𝑌
6 sspwb 4883 . . . . 5 (∅ ⊆ 𝑌 ↔ 𝒫 ∅ ⊆ 𝒫 𝑌)
75, 6mpbi 220 . . . 4 𝒫 ∅ ⊆ 𝒫 𝑌
8 ssdomg 7952 . . . 4 (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌))
94, 7, 8mpisyl 21 . . 3 (𝑋* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)
10 pweq 4138 . . . 4 (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅)
1110breq1d 4628 . . 3 (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌))
129, 11syl5ibr 236 . 2 (𝑋 = ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
13 brwdomn0 8425 . . 3 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
14 vex 3192 . . . . 5 𝑧 ∈ V
15 fopwdom 8019 . . . . 5 ((𝑧 ∈ V ∧ 𝑧:𝑌onto𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌)
1614, 15mpan 705 . . . 4 (𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1716exlimiv 1855 . . 3 (∃𝑧 𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1813, 17syl6bi 243 . 2 (𝑋 ≠ ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
1912, 18pm2.61ine 2873 1 (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wex 1701  wcel 1987  wne 2790  Vcvv 3189  wss 3559  c0 3896  𝒫 cpw 4135   class class class wbr 4618  ontowfo 5850  cdom 7904  * cwdom 8413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-dom 7908  df-wdom 8415
This theorem is referenced by:  isfin32i  9138  hsmexlem1  9199  hsmexlem3  9201  gchhar  9452
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