| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fpmd | Structured version Visualization version GIF version | ||
| Description: A total function is a partial function. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| Ref | Expression |
|---|---|
| fpmd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| fpmd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
| fpmd.c | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
| fpmd.f | ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) |
| Ref | Expression |
|---|---|
| fpmd | ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑pm 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fpmd.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 2 | fpmd.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 3 | fpmd.f | . 2 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) | |
| 4 | fpmd.c | . 2 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
| 5 | elpm2r 8885 | . 2 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) ∧ (𝐹:𝐶⟶𝐵 ∧ 𝐶 ⊆ 𝐴)) → 𝐹 ∈ (𝐵 ↑pm 𝐴)) | |
| 6 | 1, 2, 3, 4, 5 | syl22anc 839 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑pm 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ⊆ wss 3951 ⟶wf 6557 (class class class)co 7431 ↑pm cpm 8867 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pm 8869 |
| This theorem is referenced by: xlimbr 45842 fuzxrpmcn 45843 |
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