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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 4515 | . . 3 ⊢ (𝐴 = ∅ → 𝒫 𝐴 = 𝒫 ∅) | |
2 | 1 | breq1d 5049 | . 2 ⊢ (𝐴 = ∅ → (𝒫 𝐴 ≼ 𝒫 𝐵 ↔ 𝒫 ∅ ≼ 𝒫 𝐵)) |
3 | reldom 8610 | . . . . . . 7 ⊢ Rel ≼ | |
4 | 3 | brrelex1i 5590 | . . . . . 6 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
5 | 0sdomg 8753 | . . . . . 6 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝐴 ≼ 𝐵 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
7 | 6 | biimpar 481 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → ∅ ≺ 𝐴) |
8 | simpl 486 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐴 ≼ 𝐵) | |
9 | fodomr 8775 | . . . 4 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ 𝐵) → ∃𝑓 𝑓:𝐵–onto→𝐴) | |
10 | 7, 8, 9 | syl2anc 587 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:𝐵–onto→𝐴) |
11 | vex 3402 | . . . . 5 ⊢ 𝑓 ∈ V | |
12 | fopwdom 8731 | . . . . 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 1938 | . . 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 5591 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V) |
17 | 16 | pwexd 5257 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝒫 𝐵 ∈ V) |
18 | 0ss 4297 | . . . 4 ⊢ ∅ ⊆ 𝐵 | |
19 | 18 | sspwi 4513 | . . 3 ⊢ 𝒫 ∅ ⊆ 𝒫 𝐵 |
20 | ssdomg 8652 | . . 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 3017 | 1 ⊢ (𝐴 ≼ 𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ≠ wne 2932 Vcvv 3398 ⊆ wss 3853 ∅c0 4223 𝒫 cpw 4499 class class class wbr 5039 –onto→wfo 6356 ≼ cdom 8602 ≺ csdm 8603 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 |
This theorem is referenced by: djulepw 9771 gchpwdom 10249 gchaclem 10257 2ndcredom 22301 |
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