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Mirrors > Home > MPE Home > Th. List > elpm2r | Structured version Visualization version GIF version |
Description: Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.) |
Ref | Expression |
---|---|
elpm2r | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐹 ∈ (𝐴 ↑pm 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdm 6515 | . . . . . . 7 ⊢ (𝐹:𝐶⟶𝐴 → dom 𝐹 = 𝐶) | |
2 | 1 | feq2d 6493 | . . . . . 6 ⊢ (𝐹:𝐶⟶𝐴 → (𝐹:dom 𝐹⟶𝐴 ↔ 𝐹:𝐶⟶𝐴)) |
3 | 1 | sseq1d 3995 | . . . . . 6 ⊢ (𝐹:𝐶⟶𝐴 → (dom 𝐹 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐵)) |
4 | 2, 3 | anbi12d 630 | . . . . 5 ⊢ (𝐹:𝐶⟶𝐴 → ((𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵) ↔ (𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵))) |
5 | 4 | adantr 481 | . . . 4 ⊢ ((𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵) → ((𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵) ↔ (𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵))) |
6 | 5 | ibir 269 | . . 3 ⊢ ((𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)) |
7 | elpm2g 8412 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵))) | |
8 | 6, 7 | syl5ibr 247 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵) → 𝐹 ∈ (𝐴 ↑pm 𝐵))) |
9 | 8 | imp 407 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐹 ∈ (𝐴 ↑pm 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3933 dom cdm 5548 ⟶wf 6344 (class class class)co 7145 ↑pm cpm 8396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-pm 8398 |
This theorem is referenced by: fpmg 8421 pmresg 8423 rlim 14840 ello12 14861 elo12 14872 sscpwex 17073 catcfuccl 17357 catcxpccl 17445 lmbrf 21796 cnextfval 22598 lmmbrf 23792 iscauf 23810 caucfil 23813 cmetcaulem 23818 lmclimf 23834 ismbf 24156 ismbfcn 24157 mbfconst 24161 cncombf 24186 cnmbf 24187 limcfval 24397 dvfval 24422 dvnff 24447 dvn2bss 24454 dvnfre 24476 taylfvallem1 24872 taylfval 24874 tayl0 24877 taylplem1 24878 taylply2 24883 taylply 24884 dvtaylp 24885 dvntaylp 24886 dvntaylp0 24887 taylthlem1 24888 taylthlem2 24889 ulmval 24895 ulmpm 24898 iscgrgd 26226 esumcvg 31244 mrsubfval 32652 elmrsubrn 32664 msubfval 32668 fwddifval 33520 fwddifnval 33521 fpmd 41414 xlimmnfvlem2 41990 xlimpnfvlem2 41994 dvnmptdivc 42099 dvnxpaek 42103 etransclem46 42442 issmflem 42881 fdivpm 44531 refdivpm 44532 elbigo2 44540 |
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