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Theorem wdompwdom 8648
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 8636 . . . . . 6 Rel ≼*
21brrelex2i 5316 . . . . 5 (𝑋* 𝑌𝑌 ∈ V)
3 pwexg 4999 . . . . 5 (𝑌 ∈ V → 𝒫 𝑌 ∈ V)
42, 3syl 17 . . . 4 (𝑋* 𝑌 → 𝒫 𝑌 ∈ V)
5 0ss 4115 . . . . 5 ∅ ⊆ 𝑌
6 sspwb 5066 . . . . 5 (∅ ⊆ 𝑌 ↔ 𝒫 ∅ ⊆ 𝒫 𝑌)
75, 6mpbi 220 . . . 4 𝒫 ∅ ⊆ 𝒫 𝑌
8 ssdomg 8167 . . . 4 (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌))
94, 7, 8mpisyl 21 . . 3 (𝑋* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)
10 pweq 4305 . . . 4 (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅)
1110breq1d 4814 . . 3 (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌))
129, 11syl5ibr 236 . 2 (𝑋 = ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
13 brwdomn0 8639 . . 3 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
14 vex 3343 . . . . 5 𝑧 ∈ V
15 fopwdom 8233 . . . . 5 ((𝑧 ∈ V ∧ 𝑧:𝑌onto𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌)
1614, 15mpan 708 . . . 4 (𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1716exlimiv 2007 . . 3 (∃𝑧 𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1813, 17syl6bi 243 . 2 (𝑋 ≠ ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
1912, 18pm2.61ine 3015 1 (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wex 1853  wcel 2139  wne 2932  Vcvv 3340  wss 3715  c0 4058  𝒫 cpw 4302   class class class wbr 4804  ontowfo 6047  cdom 8119  * cwdom 8627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-dom 8123  df-wdom 8629
This theorem is referenced by:  isfin32i  9379  hsmexlem1  9440  hsmexlem3  9442  gchhar  9693
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