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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 4219 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
5 | 1, 2, 3, 3, 4 | offn 7683 | . 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 7681 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑥) = (𝐵𝑅𝐶)) |
10 | offveq.7 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥)) | |
11 | 9, 10 | eqtrd 2773 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑥) = (𝐻‘𝑥)) |
12 | 5, 6, 11 | eqfnfvd 7036 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Fn wfn 6539 ‘cfv 6544 (class class class)co 7409 ∘f cof 7668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 |
This theorem is referenced by: caofid0l 7701 caofid0r 7702 caofid1 7703 caofid2 7704 ofnegsub 12210 bddibl 25357 dvaddf 25459 plydivlem3 25808 poimirlem5 36493 poimirlem10 36498 poimirlem22 36510 fsuppssind 41165 ofsubid 43083 ofmul12 43084 ofdivrec 43085 ofdivcan4 43086 ofdivdiv2 43087 |
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