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Theorem wdompwdom 9513
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 9501 . . . . . 6 Rel ≼*
21brrelex2i 5689 . . . . 5 (𝑋* 𝑌𝑌 ∈ V)
32pwexd 5334 . . . 4 (𝑋* 𝑌 → 𝒫 𝑌 ∈ V)
4 0ss 4356 . . . . 5 ∅ ⊆ 𝑌
54sspwi 4572 . . . 4 𝒫 ∅ ⊆ 𝒫 𝑌
6 ssdomg 8939 . . . 4 (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌))
73, 5, 6mpisyl 21 . . 3 (𝑋* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)
8 pweq 4574 . . . 4 (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅)
98breq1d 5115 . . 3 (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌))
107, 9syl5ibr 245 . 2 (𝑋 = ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
11 brwdomn0 9504 . . 3 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
12 vex 3449 . . . . 5 𝑧 ∈ V
13 fopwdom 9023 . . . . 5 ((𝑧 ∈ V ∧ 𝑧:𝑌onto𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌)
1412, 13mpan 688 . . . 4 (𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1514exlimiv 1933 . . 3 (∃𝑧 𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1611, 15syl6bi 252 . 2 (𝑋 ≠ ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
1710, 16pm2.61ine 3028 1 (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wex 1781  wcel 2106  wne 2943  Vcvv 3445  wss 3910  c0 4282  𝒫 cpw 4560   class class class wbr 5105  ontowfo 6494  cdom 8880  * cwdom 9499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-dom 8884  df-wdom 9500
This theorem is referenced by:  isfin32i  10300  hsmexlem1  10361  hsmexlem3  10363  gchhar  10614
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