![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4419 | . . 3 ⊢ (𝐴 = ∅ → 𝒫 𝐴 = 𝒫 ∅) | |
2 | 1 | breq1d 4933 | . 2 ⊢ (𝐴 = ∅ → (𝒫 𝐴 ≼ 𝒫 𝐵 ↔ 𝒫 ∅ ≼ 𝒫 𝐵)) |
3 | reldom 8304 | . . . . . . 7 ⊢ Rel ≼ | |
4 | 3 | brrelex1i 5451 | . . . . . 6 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
5 | 0sdomg 8434 | . . . . . 6 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝐴 ≼ 𝐵 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
7 | 6 | biimpar 470 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → ∅ ≺ 𝐴) |
8 | simpl 475 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐴 ≼ 𝐵) | |
9 | fodomr 8456 | . . . 4 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ 𝐵) → ∃𝑓 𝑓:𝐵–onto→𝐴) | |
10 | 7, 8, 9 | syl2anc 576 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:𝐵–onto→𝐴) |
11 | vex 3412 | . . . . 5 ⊢ 𝑓 ∈ V | |
12 | fopwdom 8413 | . . . . 5 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐵–onto→𝐴) → 𝒫 𝐴 ≼ 𝒫 𝐵) | |
13 | 11, 12 | mpan 677 | . . . 4 ⊢ (𝑓:𝐵–onto→𝐴 → 𝒫 𝐴 ≼ 𝒫 𝐵) |
14 | 13 | exlimiv 1889 | . . 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 5452 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V) |
17 | 16 | pwexd 5127 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝒫 𝐵 ∈ V) |
18 | 0ss 4230 | . . . 4 ⊢ ∅ ⊆ 𝐵 | |
19 | sspwb 5191 | . . . 4 ⊢ (∅ ⊆ 𝐵 ↔ 𝒫 ∅ ⊆ 𝒫 𝐵) | |
20 | 18, 19 | mpbi 222 | . . 3 ⊢ 𝒫 ∅ ⊆ 𝒫 𝐵 |
21 | ssdomg 8344 | . . 3 ⊢ (𝒫 𝐵 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝐵 → 𝒫 ∅ ≼ 𝒫 𝐵)) | |
22 | 17, 20, 21 | mpisyl 21 | . 2 ⊢ (𝐴 ≼ 𝐵 → 𝒫 ∅ ≼ 𝒫 𝐵) |
23 | 2, 15, 22 | pm2.61ne 3047 | 1 ⊢ (𝐴 ≼ 𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∃wex 1742 ∈ wcel 2048 ≠ wne 2961 Vcvv 3409 ⊆ wss 3825 ∅c0 4173 𝒫 cpw 4416 class class class wbr 4923 –onto→wfo 6180 ≼ cdom 8296 ≺ csdm 8297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 |
This theorem is referenced by: djulepw 9408 gchpwdom 9882 gchaclem 9890 2ndcredom 21752 |
Copyright terms: Public domain | W3C validator |