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Theorem wdompwdom 8831
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 8819 . . . . . 6 Rel ≼*
21brrelex2i 5453 . . . . 5 (𝑋* 𝑌𝑌 ∈ V)
32pwexd 5127 . . . 4 (𝑋* 𝑌 → 𝒫 𝑌 ∈ V)
4 0ss 4230 . . . . 5 ∅ ⊆ 𝑌
5 sspwb 5192 . . . . 5 (∅ ⊆ 𝑌 ↔ 𝒫 ∅ ⊆ 𝒫 𝑌)
64, 5mpbi 222 . . . 4 𝒫 ∅ ⊆ 𝒫 𝑌
7 ssdomg 8346 . . . 4 (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌))
83, 6, 7mpisyl 21 . . 3 (𝑋* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)
9 pweq 4419 . . . 4 (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅)
109breq1d 4933 . . 3 (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌))
118, 10syl5ibr 238 . 2 (𝑋 = ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
12 brwdomn0 8822 . . 3 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
13 vex 3412 . . . . 5 𝑧 ∈ V
14 fopwdom 8415 . . . . 5 ((𝑧 ∈ V ∧ 𝑧:𝑌onto𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌)
1513, 14mpan 677 . . . 4 (𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1615exlimiv 1889 . . 3 (∃𝑧 𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1712, 16syl6bi 245 . 2 (𝑋 ≠ ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
1811, 17pm2.61ine 3045 1 (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wex 1742  wcel 2050  wne 2961  Vcvv 3409  wss 3823  c0 4172  𝒫 cpw 4416   class class class wbr 4923  ontowfo 6180  cdom 8298  * cwdom 8810
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 2744  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
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 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-dom 8302  df-wdom 8812
This theorem is referenced by:  isfin32i  9579  hsmexlem1  9640  hsmexlem3  9642  gchhar  9893
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