![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 8141 | . 2 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) ∧ (𝐹:𝐶⟶𝐵 ∧ 𝐶 ⊆ 𝐴)) → 𝐹 ∈ (𝐵 ↑pm 𝐴)) | |
6 | 1, 2, 3, 4, 5 | syl22anc 874 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑pm 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 ⊆ wss 3799 ⟶wf 6120 (class class class)co 6906 ↑pm cpm 8124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-pm 8126 |
This theorem is referenced by: xlimbr 40849 fuzxrpmcn 40850 |
Copyright terms: Public domain | W3C validator |