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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 9171 | . . . . . 6 ⊢ Rel ≼* | |
2 | 1 | brrelex2i 5595 | . . . . 5 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
3 | 2 | pwexd 5261 | . . . 4 ⊢ (𝑋 ≼* 𝑌 → 𝒫 𝑌 ∈ V) |
4 | 0ss 4301 | . . . . 5 ⊢ ∅ ⊆ 𝑌 | |
5 | 4 | sspwi 4517 | . . . 4 ⊢ 𝒫 ∅ ⊆ 𝒫 𝑌 |
6 | ssdomg 8663 | . . . 4 ⊢ (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)) | |
7 | 3, 5, 6 | mpisyl 21 | . . 3 ⊢ (𝑋 ≼* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌) |
8 | pweq 4519 | . . . 4 ⊢ (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅) | |
9 | 8 | breq1d 5053 | . . 3 ⊢ (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌)) |
10 | 7, 9 | syl5ibr 249 | . 2 ⊢ (𝑋 = ∅ → (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)) |
11 | brwdomn0 9174 | . . 3 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) | |
12 | vex 3405 | . . . . 5 ⊢ 𝑧 ∈ V | |
13 | fopwdom 8742 | . . . . 5 ⊢ ((𝑧 ∈ V ∧ 𝑧:𝑌–onto→𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌) | |
14 | 12, 13 | mpan 690 | . . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
15 | 14 | exlimiv 1938 | . . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
16 | 11, 15 | syl6bi 256 | . 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)) |
17 | 10, 16 | pm2.61ine 3018 | 1 ⊢ (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∃wex 1787 ∈ wcel 2110 ≠ wne 2935 Vcvv 3401 ⊆ wss 3857 ∅c0 4227 𝒫 cpw 4503 class class class wbr 5043 –onto→wfo 6367 ≼ cdom 8613 ≼* cwdom 9169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-dom 8617 df-wdom 9170 |
This theorem is referenced by: isfin32i 9962 hsmexlem1 10023 hsmexlem3 10025 gchhar 10276 |
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