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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 4226 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
| 5 | 1, 2, 3, 3, 4 | offn 7711 | . 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 7709 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑥) = (𝐵𝑅𝐶)) | 
| 10 | offveq.7 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥)) | |
| 11 | 9, 10 | eqtrd 2776 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑥) = (𝐻‘𝑥)) | 
| 12 | 5, 6, 11 | eqfnfvd 7053 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = 𝐻) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Fn wfn 6555 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 | 
| This theorem is referenced by: caofid0l 7731 caofid0r 7732 caofid1 7733 caofid2 7734 ofnegsub 12265 psdmul 22171 bddibl 25876 dvaddf 25980 plydivlem3 26338 poimirlem5 37633 poimirlem10 37638 poimirlem22 37650 fsuppssind 42608 ofsubid 44348 ofmul12 44349 ofdivrec 44350 ofdivcan4 44351 ofdivdiv2 44352 | 
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