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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 4111 | . . 3 ⊢ (𝐴 = ∅ → 𝒫 𝐴 = 𝒫 ∅) | |
2 | 1 | breq1d 4593 | . 2 ⊢ (𝐴 = ∅ → (𝒫 𝐴 ≼ 𝒫 𝐵 ↔ 𝒫 ∅ ≼ 𝒫 𝐵)) |
3 | reldom 7847 | . . . . . . 7 ⊢ Rel ≼ | |
4 | 3 | brrelexi 5082 | . . . . . 6 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
5 | 0sdomg 7974 | . . . . . 6 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝐴 ≼ 𝐵 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
7 | 6 | biimpar 501 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → ∅ ≺ 𝐴) |
8 | simpl 472 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐴 ≼ 𝐵) | |
9 | fodomr 7996 | . . . 4 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ 𝐵) → ∃𝑓 𝑓:𝐵–onto→𝐴) | |
10 | 7, 8, 9 | syl2anc 691 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:𝐵–onto→𝐴) |
11 | vex 3176 | . . . . 5 ⊢ 𝑓 ∈ V | |
12 | fopwdom 7953 | . . . . 5 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐵–onto→𝐴) → 𝒫 𝐴 ≼ 𝒫 𝐵) | |
13 | 11, 12 | mpan 702 | . . . 4 ⊢ (𝑓:𝐵–onto→𝐴 → 𝒫 𝐴 ≼ 𝒫 𝐵) |
14 | 13 | exlimiv 1845 | . . 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 5083 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V) |
17 | pwexg 4776 | . . . 4 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝒫 𝐵 ∈ V) |
19 | 0ss 3924 | . . . 4 ⊢ ∅ ⊆ 𝐵 | |
20 | sspwb 4844 | . . . 4 ⊢ (∅ ⊆ 𝐵 ↔ 𝒫 ∅ ⊆ 𝒫 𝐵) | |
21 | 19, 20 | mpbi 219 | . . 3 ⊢ 𝒫 ∅ ⊆ 𝒫 𝐵 |
22 | ssdomg 7887 | . . 3 ⊢ (𝒫 𝐵 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝐵 → 𝒫 ∅ ≼ 𝒫 𝐵)) | |
23 | 18, 21, 22 | mpisyl 21 | . 2 ⊢ (𝐴 ≼ 𝐵 → 𝒫 ∅ ≼ 𝒫 𝐵) |
24 | 2, 15, 23 | pm2.61ne 2867 | 1 ⊢ (𝐴 ≼ 𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 class class class wbr 4583 –onto→wfo 5802 ≼ cdom 7839 ≺ csdm 7840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 |
This theorem is referenced by: cdalepw 8901 gchpwdom 9371 gchaclem 9379 2ndcredom 21063 |
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