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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 4619 | . . 3 ⊢ (𝐴 = ∅ → 𝒫 𝐴 = 𝒫 ∅) | |
2 | 1 | breq1d 5158 | . 2 ⊢ (𝐴 = ∅ → (𝒫 𝐴 ≼ 𝒫 𝐵 ↔ 𝒫 ∅ ≼ 𝒫 𝐵)) |
3 | reldom 8990 | . . . . . . 7 ⊢ Rel ≼ | |
4 | 3 | brrelex1i 5745 | . . . . . 6 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
5 | 0sdomg 9143 | . . . . . 6 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝐴 ≼ 𝐵 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
7 | 6 | biimpar 477 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → ∅ ≺ 𝐴) |
8 | simpl 482 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐴 ≼ 𝐵) | |
9 | fodomr 9167 | . . . 4 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ 𝐵) → ∃𝑓 𝑓:𝐵–onto→𝐴) | |
10 | 7, 8, 9 | syl2anc 584 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:𝐵–onto→𝐴) |
11 | vex 3482 | . . . . 5 ⊢ 𝑓 ∈ V | |
12 | fopwdom 9119 | . . . . 5 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐵–onto→𝐴) → 𝒫 𝐴 ≼ 𝒫 𝐵) | |
13 | 11, 12 | mpan 690 | . . . 4 ⊢ (𝑓:𝐵–onto→𝐴 → 𝒫 𝐴 ≼ 𝒫 𝐵) |
14 | 13 | exlimiv 1928 | . . 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 5746 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V) |
17 | 16 | pwexd 5385 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝒫 𝐵 ∈ V) |
18 | 0ss 4406 | . . . 4 ⊢ ∅ ⊆ 𝐵 | |
19 | 18 | sspwi 4617 | . . 3 ⊢ 𝒫 ∅ ⊆ 𝒫 𝐵 |
20 | ssdomg 9039 | . . 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 3025 | 1 ⊢ (𝐴 ≼ 𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 class class class wbr 5148 –onto→wfo 6561 ≼ cdom 8982 ≺ csdm 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-en 8985 df-dom 8986 df-sdom 8987 |
This theorem is referenced by: djulepw 10231 gchpwdom 10708 gchaclem 10716 2ndcredom 23474 |
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