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Theorem wdompwdom 8366
 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 8354 . . . . . 6 Rel ≼*
21brrelex2i 5083 . . . . 5 (𝑋* 𝑌𝑌 ∈ V)
3 pwexg 4776 . . . . 5 (𝑌 ∈ V → 𝒫 𝑌 ∈ V)
42, 3syl 17 . . . 4 (𝑋* 𝑌 → 𝒫 𝑌 ∈ V)
5 0ss 3924 . . . . 5 ∅ ⊆ 𝑌
6 sspwb 4844 . . . . 5 (∅ ⊆ 𝑌 ↔ 𝒫 ∅ ⊆ 𝒫 𝑌)
75, 6mpbi 219 . . . 4 𝒫 ∅ ⊆ 𝒫 𝑌
8 ssdomg 7887 . . . 4 (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌))
94, 7, 8mpisyl 21 . . 3 (𝑋* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)
10 pweq 4111 . . . 4 (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅)
1110breq1d 4593 . . 3 (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌))
129, 11syl5ibr 235 . 2 (𝑋 = ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
13 brwdomn0 8357 . . 3 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
14 vex 3176 . . . . 5 𝑧 ∈ V
15 fopwdom 7953 . . . . 5 ((𝑧 ∈ V ∧ 𝑧:𝑌onto𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌)
1614, 15mpan 702 . . . 4 (𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1716exlimiv 1845 . . 3 (∃𝑧 𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1813, 17syl6bi 242 . 2 (𝑋 ≠ ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
1912, 18pm2.61ine 2865 1 (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108   class class class wbr 4583  –onto→wfo 5802   ≼ cdom 7839   ≼* cwdom 8345 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-dom 7843  df-wdom 8347 This theorem is referenced by:  isfin32i  9070  hsmexlem1  9131  hsmexlem3  9133  gchhar  9380
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