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Mirrors > Home > ILE Home > Th. List > elpmi | 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 | df-pm 6553 | . . . 4 ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}) | |
2 | 1 | elmpocl 5976 | . . 3 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | elpm2g 6567 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵))) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵))) |
5 | 4 | ibi 175 | 1 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1481 {crab 2421 Vcvv 2689 ⊆ wss 3076 𝒫 cpw 3515 × cxp 4545 dom cdm 4547 Fun wfun 5125 ⟶wf 5127 (class class class)co 5782 ↑pm cpm 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pm 6553 |
This theorem is referenced by: pmfun 6570 pmresg 6578 ennnfonelemg 11952 ennnfonelemf1 11967 reldvg 12856 dvbsssg 12863 dvfgg 12865 |
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