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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 6745 | . . . . . . 7 ⊢ (𝐹:𝐶⟶𝐴 → dom 𝐹 = 𝐶) | |
| 2 | 1 | feq2d 6722 | . . . . . 6 ⊢ (𝐹:𝐶⟶𝐴 → (𝐹:dom 𝐹⟶𝐴 ↔ 𝐹:𝐶⟶𝐴)) |
| 3 | 1 | sseq1d 4015 | . . . . . 6 ⊢ (𝐹:𝐶⟶𝐴 → (dom 𝐹 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐵)) |
| 4 | 2, 3 | anbi12d 632 | . . . . 5 ⊢ (𝐹:𝐶⟶𝐴 → ((𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵) ↔ (𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵))) |
| 5 | 4 | adantr 480 | . . . 4 ⊢ ((𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵) → ((𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵) ↔ (𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵))) |
| 6 | 5 | ibir 268 | . . 3 ⊢ ((𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)) |
| 7 | elpm2g 8884 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵))) | |
| 8 | 6, 7 | imbitrrid 246 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵) → 𝐹 ∈ (𝐴 ↑pm 𝐵))) |
| 9 | 8 | imp 406 | 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 206 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 dom cdm 5685 ⟶wf 6557 (class class class)co 7431 ↑pm cpm 8867 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pm 8869 |
| This theorem is referenced by: fpmg 8908 pmresg 8910 rlim 15531 ello12 15552 elo12 15563 sscpwex 17859 catcfuccl 18163 catcxpccl 18252 lmbrf 23268 cnextfval 24070 lmmbrf 25296 iscauf 25314 caucfil 25317 cmetcaulem 25322 lmclimf 25338 ismbf 25663 ismbfcn 25664 mbfconst 25668 cncombf 25693 cnmbf 25694 limcfval 25907 dvfval 25932 dvnff 25959 dvn2bss 25966 dvnfre 25990 taylfvallem1 26398 taylfval 26400 tayl0 26403 taylplem1 26404 taylply2 26409 taylply2OLD 26410 taylply 26411 dvtaylp 26412 dvntaylp 26413 dvntaylp0 26414 taylthlem1 26415 taylthlem2 26416 taylthlem2OLD 26417 ulmval 26423 ulmpm 26426 iscgrgd 28521 esumcvg 34087 mrsubfval 35513 elmrsubrn 35525 msubfval 35529 fwddifval 36163 fwddifnval 36164 fpmd 45270 xlimmnfvlem2 45848 xlimpnfvlem2 45852 dvnmptdivc 45953 dvnxpaek 45957 etransclem46 46295 issmflem 46742 fdivpm 48464 refdivpm 48465 elbigo2 48473 |
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