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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 4183 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
5 | 1, 2, 3, 3, 4 | offn 7635 | . 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 7633 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑥) = (𝐵𝑅𝐶)) |
10 | offveq.7 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥)) | |
11 | 9, 10 | eqtrd 2777 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑥) = (𝐻‘𝑥)) |
12 | 5, 6, 11 | eqfnfvd 6990 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Fn wfn 6496 ‘cfv 6501 (class class class)co 7362 ∘f cof 7620 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 |
This theorem is referenced by: caofid0l 7653 caofid0r 7654 caofid1 7655 caofid2 7656 ofnegsub 12158 bddibl 25220 dvaddf 25322 plydivlem3 25671 poimirlem5 36112 poimirlem10 36117 poimirlem22 36129 fsuppssind 40797 ofsubid 42678 ofmul12 42679 ofdivrec 42680 ofdivcan4 42681 ofdivdiv2 42682 |
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