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Theorem wdompwdom 9575
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 9563 . . . . . 6 Rel ≼*
21brrelex2i 5726 . . . . 5 (𝑋* 𝑌𝑌 ∈ V)
32pwexd 5370 . . . 4 (𝑋* 𝑌 → 𝒫 𝑌 ∈ V)
4 0ss 4391 . . . . 5 ∅ ⊆ 𝑌
54sspwi 4609 . . . 4 𝒫 ∅ ⊆ 𝒫 𝑌
6 ssdomg 8998 . . . 4 (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌))
73, 5, 6mpisyl 21 . . 3 (𝑋* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)
8 pweq 4611 . . . 4 (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅)
98breq1d 5151 . . 3 (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌))
107, 9imbitrrid 245 . 2 (𝑋 = ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
11 brwdomn0 9566 . . 3 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
12 vex 3472 . . . . 5 𝑧 ∈ V
13 fopwdom 9082 . . . . 5 ((𝑧 ∈ V ∧ 𝑧:𝑌onto𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌)
1412, 13mpan 687 . . . 4 (𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1514exlimiv 1925 . . 3 (∃𝑧 𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1611, 15biimtrdi 252 . 2 (𝑋 ≠ ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
1710, 16pm2.61ine 3019 1 (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wex 1773  wcel 2098  wne 2934  Vcvv 3468  wss 3943  c0 4317  𝒫 cpw 4597   class class class wbr 5141  ontowfo 6535  cdom 8939  * cwdom 9561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-dom 8943  df-wdom 9562
This theorem is referenced by:  isfin32i  10362  hsmexlem1  10423  hsmexlem3  10425  gchhar  10676
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