Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ofcfeqd2 | Structured version Visualization version GIF version |
Description: Equality theorem for function/constant operation value. (Contributed by Thierry Arnoux, 31-Jan-2017.) |
Ref | Expression |
---|---|
ofcfeqd2.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
ofcfeqd2.2 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶)) |
ofcfeqd2.3 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
ofcfeqd2.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ofcfeqd2.5 | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
Ref | Expression |
---|---|
ofcfeqd2 | ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f/c 𝑃𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7165 | . . . . 5 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦𝑅𝐶) = ((𝐹‘𝑥)𝑅𝐶)) | |
2 | oveq1 7165 | . . . . 5 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦𝑃𝐶) = ((𝐹‘𝑥)𝑃𝐶)) | |
3 | 1, 2 | eqeq12d 2839 | . . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝑦𝑅𝐶) = (𝑦𝑃𝐶) ↔ ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑥)𝑃𝐶))) |
4 | ofcfeqd2.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶)) | |
5 | 4 | ralrimiva 3184 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶)) |
6 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶)) |
7 | ofcfeqd2.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) | |
8 | 3, 6, 7 | rspcdva 3627 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑥)𝑃𝐶)) |
9 | 8 | mpteq2dva 5163 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑃𝐶))) |
10 | ofcfeqd2.3 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
11 | ofcfeqd2.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
12 | ofcfeqd2.5 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
13 | eqidd 2824 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
14 | 10, 11, 12, 13 | ofcfval 31359 | . 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
15 | 10, 11, 12, 13 | ofcfval 31359 | . 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑃𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑃𝐶))) |
16 | 9, 14, 15 | 3eqtr4d 2868 | 1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f/c 𝑃𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ↦ cmpt 5148 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 ∘f/c cofc 31356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-ofc 31357 |
This theorem is referenced by: coinfliplem 31738 |
Copyright terms: Public domain | W3C validator |