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Mirrors > Home > MPE Home > Th. List > wdompwdom | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
wdompwdom | ⊢ (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relwdom 9589 | . . . . . 6 ⊢ Rel ≼* | |
2 | 1 | brrelex2i 5729 | . . . . 5 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
3 | 2 | pwexd 5373 | . . . 4 ⊢ (𝑋 ≼* 𝑌 → 𝒫 𝑌 ∈ V) |
4 | 0ss 4392 | . . . . 5 ⊢ ∅ ⊆ 𝑌 | |
5 | 4 | sspwi 4610 | . . . 4 ⊢ 𝒫 ∅ ⊆ 𝒫 𝑌 |
6 | ssdomg 9019 | . . . 4 ⊢ (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)) | |
7 | 3, 5, 6 | mpisyl 21 | . . 3 ⊢ (𝑋 ≼* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌) |
8 | pweq 4612 | . . . 4 ⊢ (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅) | |
9 | 8 | breq1d 5153 | . . 3 ⊢ (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌)) |
10 | 7, 9 | imbitrrid 245 | . 2 ⊢ (𝑋 = ∅ → (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)) |
11 | brwdomn0 9592 | . . 3 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) | |
12 | vex 3467 | . . . . 5 ⊢ 𝑧 ∈ V | |
13 | fopwdom 9103 | . . . . 5 ⊢ ((𝑧 ∈ V ∧ 𝑧:𝑌–onto→𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌) | |
14 | 12, 13 | mpan 688 | . . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
15 | 14 | exlimiv 1925 | . . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
16 | 11, 15 | biimtrdi 252 | . 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)) |
17 | 10, 16 | pm2.61ine 3015 | 1 ⊢ (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2930 Vcvv 3463 ⊆ wss 3939 ∅c0 4318 𝒫 cpw 4598 class class class wbr 5143 –onto→wfo 6541 ≼ cdom 8960 ≼* cwdom 9587 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-dom 8964 df-wdom 9588 |
This theorem is referenced by: isfin32i 10388 hsmexlem1 10449 hsmexlem3 10451 gchhar 10702 |
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