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Mirrors > Home > MPE Home > Th. List > elpm2 | Structured version Visualization version GIF version |
Description: The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.) |
Ref | Expression |
---|---|
elmap.1 | ⊢ 𝐴 ∈ V |
elmap.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
elpm2 | ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmap.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | elmap.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | elpm2g 7874 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵))) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 dom cdm 5114 ⟶wf 5884 (class class class)co 6650 ↑pm cpm 7858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-pm 7860 |
This theorem is referenced by: rlimf 14232 rlimss 14233 lo1f 14249 lo1dm 14250 o1f 14260 o1dm 14261 coapm 16721 pmltpclem2 23218 mbff 23394 limcrcl 23638 dvnres 23694 c1liplem1 23759 c1lip2 23761 ulmf2 24138 elbigof 42348 elbigodm 42349 elbigoimp 42350 |
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