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Mirrors > Home > MPE Home > Th. List > offun | Structured version Visualization version GIF version |
Description: The function operation produces a function. (Contributed by SN, 23-Jul-2024.) |
Ref | Expression |
---|---|
offun.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
offun.2 | ⊢ (𝜑 → 𝐺 Fn 𝐵) |
offun.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
offun.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
offun | ⊢ (𝜑 → Fun (𝐹 ∘f 𝑅𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offun.1 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | offun.2 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
3 | offun.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | offun.4 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
5 | eqid 2758 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐵) | |
6 | 1, 2, 3, 4, 5 | offn 7423 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn (𝐴 ∩ 𝐵)) |
7 | fnfun 6439 | . 2 ⊢ ((𝐹 ∘f 𝑅𝐺) Fn (𝐴 ∩ 𝐵) → Fun (𝐹 ∘f 𝑅𝐺)) | |
8 | 6, 7 | syl 17 | 1 ⊢ (𝜑 → Fun (𝐹 ∘f 𝑅𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ∩ cin 3859 Fun wfun 6334 Fn wfn 6335 (class class class)co 7156 ∘f cof 7409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7411 |
This theorem is referenced by: lcomfsupp 19755 frlmsslsp 20574 psrbagev1 20851 psrbagev1OLD 20852 mhpmulcl 20905 mndpsuppss 45189 mndpfsupp 45194 |
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