Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > offn | Structured version Visualization version GIF version |
Description: The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
Ref | Expression |
---|---|
offval.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
offval.2 | ⊢ (𝜑 → 𝐺 Fn 𝐵) |
offval.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
offval.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
offval.5 | ⊢ (𝐴 ∩ 𝐵) = 𝑆 |
Ref | Expression |
---|---|
offn | ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7191 | . . 3 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V | |
2 | eqid 2823 | . . 3 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) | |
3 | 1, 2 | fnmpti 6493 | . 2 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn 𝑆 |
4 | offval.1 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
5 | offval.2 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
6 | offval.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | offval.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
8 | offval.5 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = 𝑆 | |
9 | eqidd 2824 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
10 | eqidd 2824 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
11 | 4, 5, 6, 7, 8, 9, 10 | offval 7418 | . . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
12 | 11 | fneq1d 6448 | . 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) Fn 𝑆 ↔ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn 𝑆)) |
13 | 3, 12 | mpbiri 260 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ↦ cmpt 5148 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 |
This theorem is referenced by: offveq 7432 suppofss1d 7870 suppofss2d 7871 ofsubeq0 11637 ofnegsub 11638 ofsubge0 11639 seqof 13430 ofccat 14331 lcomfsupp 19676 psrbagcon 20153 psrbagev1 20292 frlmsslsp 20942 frlmup1 20944 i1faddlem 24296 i1fmullem 24297 dv11cn 24600 coemulc 24847 ofmulrt 24873 plydivlem3 24886 plyrem 24896 jensen 25568 basellem9 25668 broucube 34928 caofcan 40662 ofmul12 40664 ofdivrec 40665 ofdivcan4 40666 ofdivdiv2 40667 mndpsuppss 44426 mndpfsupp 44431 |
Copyright terms: Public domain | W3C validator |