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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 3879 | . . . . . . 7 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → ¬ (𝐴 ↑pm 𝐵) = ∅) | |
2 | fnpm 7752 | . . . . . . . . 9 ⊢ ↑pm Fn (V × V) | |
3 | fndm 5904 | . . . . . . . . 9 ⊢ ( ↑pm Fn (V × V) → dom ↑pm = (V × V)) | |
4 | 2, 3 | ax-mp 5 | . . . . . . . 8 ⊢ dom ↑pm = (V × V) |
5 | 4 | ndmov 6716 | . . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑pm 𝐵) = ∅) |
6 | 1, 5 | nsyl2 141 | . . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | elpmg 7759 | . . . . . 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 255 | . . . 4 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴))) |
10 | 9 | simprd 478 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → 𝑓 ⊆ (𝐵 × 𝐴)) |
11 | selpw 4115 | . . 3 ⊢ (𝑓 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑓 ⊆ (𝐵 × 𝐴)) | |
12 | 10, 11 | sylibr 223 | . 2 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → 𝑓 ∈ 𝒫 (𝐵 × 𝐴)) |
13 | 12 | ssriv 3572 | 1 ⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 × cxp 5036 dom cdm 5038 Fun wfun 5798 Fn wfn 5799 (class class class)co 6549 ↑pm cpm 7745 |
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-iun 4457 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-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-pm 7747 |
This theorem is referenced by: mapsspw 7779 wunpm 9426 |
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