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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 5964 | . . . . . . 7 ⊢ (𝐹:𝐶⟶𝐴 → dom 𝐹 = 𝐶) | |
2 | 1 | feq2d 5944 | . . . . . 6 ⊢ (𝐹:𝐶⟶𝐴 → (𝐹:dom 𝐹⟶𝐴 ↔ 𝐹:𝐶⟶𝐴)) |
3 | 1 | sseq1d 3595 | . . . . . 6 ⊢ (𝐹:𝐶⟶𝐴 → (dom 𝐹 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐵)) |
4 | 2, 3 | anbi12d 743 | . . . . 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 256 | . . 3 ⊢ ((𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)) |
7 | elpm2g 7760 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵))) | |
8 | 6, 7 | syl5ibr 235 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵) → 𝐹 ∈ (𝐴 ↑pm 𝐵))) |
9 | 8 | imp 444 | 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 195 ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 dom cdm 5038 ⟶wf 5800 (class class class)co 6549 ↑pm cpm 7745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-pm 7747 |
This theorem is referenced by: fpmg 7769 pmresg 7771 rlim 14074 ello12 14095 elo12 14106 sscpwex 16298 catcfuccl 16582 catcxpccl 16670 lmbrf 20874 cnextfval 21676 lmmbrf 22868 iscauf 22886 caucfil 22889 cmetcaulem 22894 lmclimf 22910 ismbf 23203 ismbfcn 23204 mbfconst 23208 cncombf 23231 cnmbf 23232 limcfval 23442 dvfval 23467 dvnff 23492 dvn2bss 23499 dvnfre 23521 taylfvallem1 23915 taylfval 23917 tayl0 23920 taylplem1 23921 taylply2 23926 taylply 23927 dvtaylp 23928 dvntaylp 23929 dvntaylp0 23930 taylthlem1 23931 taylthlem2 23932 ulmval 23938 ulmpm 23941 iscgrgd 25208 iseupa 26492 esumcvg 29475 mrsubfval 30659 elmrsubrn 30671 msubfval 30675 fwddifval 31439 fwddifnval 31440 dvnmptdivc 38828 dvnxpaek 38832 etransclem46 39173 issmflem 39613 fdivpm 42135 refdivpm 42136 elbigo2 42144 |
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