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Mirrors > Home > MPE Home > Th. List > pwdom | Structured version Visualization version GIF version |
Description: Injection of sets implies injection on power sets. (Contributed by Mario Carneiro, 9-Apr-2015.) |
Ref | Expression |
---|---|
pwdom | ⊢ (𝐴 ≼ 𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 4557 | . . 3 ⊢ (𝐴 = ∅ → 𝒫 𝐴 = 𝒫 ∅) | |
2 | 1 | breq1d 5078 | . 2 ⊢ (𝐴 = ∅ → (𝒫 𝐴 ≼ 𝒫 𝐵 ↔ 𝒫 ∅ ≼ 𝒫 𝐵)) |
3 | reldom 8517 | . . . . . . 7 ⊢ Rel ≼ | |
4 | 3 | brrelex1i 5610 | . . . . . 6 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
5 | 0sdomg 8648 | . . . . . 6 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝐴 ≼ 𝐵 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
7 | 6 | biimpar 480 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → ∅ ≺ 𝐴) |
8 | simpl 485 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐴 ≼ 𝐵) | |
9 | fodomr 8670 | . . . 4 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ 𝐵) → ∃𝑓 𝑓:𝐵–onto→𝐴) | |
10 | 7, 8, 9 | syl2anc 586 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:𝐵–onto→𝐴) |
11 | vex 3499 | . . . . 5 ⊢ 𝑓 ∈ V | |
12 | fopwdom 8627 | . . . . 5 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐵–onto→𝐴) → 𝒫 𝐴 ≼ 𝒫 𝐵) | |
13 | 11, 12 | mpan 688 | . . . 4 ⊢ (𝑓:𝐵–onto→𝐴 → 𝒫 𝐴 ≼ 𝒫 𝐵) |
14 | 13 | exlimiv 1931 | . . 3 ⊢ (∃𝑓 𝑓:𝐵–onto→𝐴 → 𝒫 𝐴 ≼ 𝒫 𝐵) |
15 | 10, 14 | syl 17 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → 𝒫 𝐴 ≼ 𝒫 𝐵) |
16 | 3 | brrelex2i 5611 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V) |
17 | 16 | pwexd 5282 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝒫 𝐵 ∈ V) |
18 | 0ss 4352 | . . . 4 ⊢ ∅ ⊆ 𝐵 | |
19 | 18 | sspwi 4555 | . . 3 ⊢ 𝒫 ∅ ⊆ 𝒫 𝐵 |
20 | ssdomg 8557 | . . 3 ⊢ (𝒫 𝐵 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝐵 → 𝒫 ∅ ≼ 𝒫 𝐵)) | |
21 | 17, 19, 20 | mpisyl 21 | . 2 ⊢ (𝐴 ≼ 𝐵 → 𝒫 ∅ ≼ 𝒫 𝐵) |
22 | 2, 15, 21 | pm2.61ne 3104 | 1 ⊢ (𝐴 ≼ 𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 class class class wbr 5068 –onto→wfo 6355 ≼ cdom 8509 ≺ csdm 8510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 |
This theorem is referenced by: djulepw 9620 gchpwdom 10094 gchaclem 10102 2ndcredom 22060 |
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