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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 3784 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
5 | 1, 2, 3, 3, 4 | offn 6806 | . 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 6804 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑥) = (𝐵𝑅𝐶)) |
10 | offveq.7 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥)) | |
11 | 9, 10 | eqtrd 2644 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑥) = (𝐻‘𝑥)) |
12 | 5, 6, 11 | eqfnfvd 6222 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 |
This theorem is referenced by: caofid0l 6823 caofid0r 6824 caofid1 6825 caofid2 6826 ofnegsub 10895 bddibl 23412 dvaddf 23511 plydivlem3 23854 poimirlem5 32584 poimirlem10 32589 poimirlem22 32601 ofsubid 37545 ofmul12 37546 ofdivrec 37547 ofdivcan4 37548 ofdivdiv2 37549 |
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