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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 8883 | . 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 2105 ⊆ wss 3962 ⟶wf 6558 (class class class)co 7430 ↑pm cpm 8865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-pm 8867 |
This theorem is referenced by: xlimbr 45782 fuzxrpmcn 45783 |
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