![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > elpmrn | Structured version Visualization version GIF version |
Description: The range of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
elpmrn | ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → ran 𝐹 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpmi 8217 | . . 3 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)) | |
2 | 1 | simpld 487 | . 2 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → 𝐹:dom 𝐹⟶𝐴) |
3 | 2 | frnd 6345 | 1 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → ran 𝐹 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2048 ⊆ wss 3825 dom cdm 5400 ran crn 5401 ⟶wf 6178 (class class class)co 6970 ↑pm cpm 8199 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-1st 7494 df-2nd 7495 df-pm 8201 |
This theorem is referenced by: mbfpsssmf 42436 |
Copyright terms: Public domain | W3C validator |