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Mirrors > Home > MPE Home > Th. List > elpmi | Structured version Visualization version GIF version |
Description: A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.) |
Ref | Expression |
---|---|
elpmi | ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4273 | . . . 4 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → ¬ (𝐴 ↑pm 𝐵) = ∅) | |
2 | fnpm 8606 | . . . . . 6 ⊢ ↑pm Fn (V × V) | |
3 | 2 | fndmi 6535 | . . . . 5 ⊢ dom ↑pm = (V × V) |
4 | 3 | ndmov 7450 | . . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑pm 𝐵) = ∅) |
5 | 1, 4 | nsyl2 141 | . . 3 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
6 | elpm2g 8615 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵))) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵))) |
8 | 7 | ibi 266 | 1 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ⊆ wss 3892 ∅c0 4262 × cxp 5588 dom cdm 5590 ⟶wf 6428 (class class class)co 7271 ↑pm cpm 8599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-1st 7824 df-2nd 7825 df-pm 8601 |
This theorem is referenced by: pmfun 8618 pmresg 8641 equivcau 24462 dvn2bss 25092 mrsubff 33470 mrsubrn 33471 elpmrn 42730 elpmi2 42734 issmflem 44231 |
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