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Mirrors > Home > MPE Home > Th. List > fpmg | Structured version Visualization version GIF version |
Description: A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.) |
Ref | Expression |
---|---|
fpmg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 ∈ (𝐵 ↑pm 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3991 | . . . 4 ⊢ 𝐴 ⊆ 𝐴 | |
2 | elpm2r 8426 | . . . 4 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐴)) → 𝐹 ∈ (𝐵 ↑pm 𝐴)) | |
3 | 1, 2 | mpanr2 702 | . . 3 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) ∧ 𝐹:𝐴⟶𝐵) → 𝐹 ∈ (𝐵 ↑pm 𝐴)) |
4 | 3 | 3impa 1106 | . 2 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 ∈ (𝐵 ↑pm 𝐴)) |
5 | 4 | 3com12 1119 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 ∈ (𝐵 ↑pm 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 ⊆ wss 3938 ⟶wf 6353 (class class class)co 7158 ↑pm cpm 8409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-pm 8411 |
This theorem is referenced by: fpm 8441 mapsspm 8442 dvnff 24522 dvnply2 24878 0wlkonlem2 27900 |
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