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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mptelpm | Structured version Visualization version GIF version |
Description: A function in maps-to notation is a partial map . (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
mptelpm.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
mptelpm.a | ⊢ (𝜑 → 𝐴 ⊆ 𝐷) |
mptelpm.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
mptelpm.d | ⊢ (𝜑 → 𝐷 ∈ 𝑊) |
Ref | Expression |
---|---|
mptelpm | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐶 ↑pm 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptelpm.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
2 | 1 | fmpttd 7130 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
3 | eqid 2728 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3, 1 | dmmptd 6705 | . . . . . 6 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
5 | 4 | eqcomd 2734 | . . . . 5 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
6 | 5 | feq2d 6713 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶𝐶)) |
7 | 2, 6 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶𝐶) |
8 | mptelpm.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐷) | |
9 | 4, 8 | eqsstrd 4020 | . . 3 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐷) |
10 | 7, 9 | jca 510 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶𝐶 ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐷)) |
11 | mptelpm.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
12 | mptelpm.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑊) | |
13 | elpm2g 8869 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐶 ↑pm 𝐷) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶𝐶 ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐷))) | |
14 | 11, 12, 13 | syl2anc 582 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐶 ↑pm 𝐷) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶𝐶 ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐷))) |
15 | 10, 14 | mpbird 256 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐶 ↑pm 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ⊆ wss 3949 ↦ cmpt 5235 dom cdm 5682 ⟶wf 6549 (class class class)co 7426 ↑pm cpm 8852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-pm 8854 |
This theorem is referenced by: dvnmptconst 45358 dvnmul 45360 |
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