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Mirrors > Home > MPE Home > Th. List > pmsspw | Structured version Visualization version GIF version |
Description: Partial maps are a subset of the power set of the Cartesian product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
pmsspw | ⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 3920 | . . . . . . 7 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → ¬ (𝐴 ↑pm 𝐵) = ∅) | |
2 | fnpm 7865 | . . . . . . . . 9 ⊢ ↑pm Fn (V × V) | |
3 | fndm 5990 | . . . . . . . . 9 ⊢ ( ↑pm Fn (V × V) → dom ↑pm = (V × V)) | |
4 | 2, 3 | ax-mp 5 | . . . . . . . 8 ⊢ dom ↑pm = (V × V) |
5 | 4 | ndmov 6818 | . . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑pm 𝐵) = ∅) |
6 | 1, 5 | nsyl2 142 | . . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | elpmg 7873 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑓 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴)))) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (𝑓 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴)))) |
9 | 8 | ibi 256 | . . . 4 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴))) |
10 | 9 | simprd 479 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → 𝑓 ⊆ (𝐵 × 𝐴)) |
11 | selpw 4165 | . . 3 ⊢ (𝑓 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑓 ⊆ (𝐵 × 𝐴)) | |
12 | 10, 11 | sylibr 224 | . 2 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → 𝑓 ∈ 𝒫 (𝐵 × 𝐴)) |
13 | 12 | ssriv 3607 | 1 ⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 × cxp 5112 dom cdm 5114 Fun wfun 5882 Fn wfn 5883 (class class class)co 6650 ↑pm cpm 7858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-pm 7860 |
This theorem is referenced by: mapsspw 7893 wunpm 9547 |
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