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 8504 | . 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 2112 ⊆ wss 3853 ⟶wf 6354 (class class class)co 7191 ↑pm cpm 8487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-pm 8489 |
This theorem is referenced by: xlimbr 42986 fuzxrpmcn 42987 |
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