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Mirrors > Home > ILE Home > Th. List > ofexg | GIF version |
Description: A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.) |
Ref | Expression |
---|---|
ofexg | ⊢ (𝐴 ∈ 𝑉 → ( ∘𝑓 𝑅 ↾ 𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-of 6080 | . . 3 ⊢ ∘𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | |
2 | 1 | mpofun 5974 | . 2 ⊢ Fun ∘𝑓 𝑅 |
3 | resfunexg 5736 | . 2 ⊢ ((Fun ∘𝑓 𝑅 ∧ 𝐴 ∈ 𝑉) → ( ∘𝑓 𝑅 ↾ 𝐴) ∈ V) | |
4 | 2, 3 | mpan 424 | 1 ⊢ (𝐴 ∈ 𝑉 → ( ∘𝑓 𝑅 ↾ 𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 Vcvv 2737 ∩ cin 3128 ↦ cmpt 4063 dom cdm 4625 ↾ cres 4627 Fun wfun 5209 ‘cfv 5215 (class class class)co 5872 ∘𝑓 cof 6078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-oprab 5876 df-mpo 5877 df-of 6080 |
This theorem is referenced by: ofmresex 6135 |
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