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Mirrors > Home > MPE Home > Th. List > offveq | Structured version Visualization version GIF version |
Description: Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
offveq.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
offveq.2 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
offveq.3 | ⊢ (𝜑 → 𝐺 Fn 𝐴) |
offveq.4 | ⊢ (𝜑 → 𝐻 Fn 𝐴) |
offveq.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) |
offveq.6 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶) |
offveq.7 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥)) |
Ref | Expression |
---|---|
offveq | ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offveq.2 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | offveq.3 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) | |
3 | offveq.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | inidm 4217 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
5 | 1, 2, 3, 3, 4 | offn 7698 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝐴) |
6 | offveq.4 | . 2 ⊢ (𝜑 → 𝐻 Fn 𝐴) | |
7 | offveq.5 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) | |
8 | offveq.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶) | |
9 | 1, 2, 3, 3, 4, 7, 8 | ofval 7696 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑥) = (𝐵𝑅𝐶)) |
10 | offveq.7 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥)) | |
11 | 9, 10 | eqtrd 2765 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑥) = (𝐻‘𝑥)) |
12 | 5, 6, 11 | eqfnfvd 7042 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Fn wfn 6544 ‘cfv 6549 (class class class)co 7419 ∘f cof 7683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 |
This theorem is referenced by: caofid0l 7717 caofid0r 7718 caofid1 7719 caofid2 7720 ofnegsub 12243 psdmul 22113 bddibl 25813 dvaddf 25917 plydivlem3 26275 poimirlem5 37226 poimirlem10 37231 poimirlem22 37243 fsuppssind 41958 ofsubid 43900 ofmul12 43901 ofdivrec 43902 ofdivcan4 43903 ofdivdiv2 43904 |
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