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Theorem wdompwdom 9030
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 9018 . . . . . 6 Rel ≼*
21brrelex2i 5577 . . . . 5 (𝑋* 𝑌𝑌 ∈ V)
32pwexd 5248 . . . 4 (𝑋* 𝑌 → 𝒫 𝑌 ∈ V)
4 0ss 4307 . . . . 5 ∅ ⊆ 𝑌
54sspwi 4514 . . . 4 𝒫 ∅ ⊆ 𝒫 𝑌
6 ssdomg 8542 . . . 4 (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌))
73, 5, 6mpisyl 21 . . 3 (𝑋* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)
8 pweq 4516 . . . 4 (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅)
98breq1d 5043 . . 3 (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌))
107, 9syl5ibr 249 . 2 (𝑋 = ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
11 brwdomn0 9021 . . 3 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
12 vex 3447 . . . . 5 𝑧 ∈ V
13 fopwdom 8612 . . . . 5 ((𝑧 ∈ V ∧ 𝑧:𝑌onto𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌)
1412, 13mpan 689 . . . 4 (𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1514exlimiv 1931 . . 3 (∃𝑧 𝑧:𝑌onto𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌)
1611, 15syl6bi 256 . 2 (𝑋 ≠ ∅ → (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌))
1710, 16pm2.61ine 3073 1 (𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wex 1781  wcel 2112  wne 2990  Vcvv 3444  wss 3884  c0 4246  𝒫 cpw 4500   class class class wbr 5033  ontowfo 6326  cdom 8494  * cwdom 9016
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-dom 8498  df-wdom 9017
This theorem is referenced by:  isfin32i  9780  hsmexlem1  9841  hsmexlem3  9843  gchhar  10094
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