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Mathbox for Thierry Arnoux |
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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/𝑐𝑅𝐶) = (𝐹∘𝑓/𝑐𝑃𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6981 | . . . . 5 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦𝑅𝐶) = ((𝐹‘𝑥)𝑅𝐶)) | |
2 | oveq1 6981 | . . . . 5 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦𝑃𝐶) = ((𝐹‘𝑥)𝑃𝐶)) | |
3 | 1, 2 | eqeq12d 2786 | . . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝑦𝑅𝐶) = (𝑦𝑃𝐶) ↔ ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑥)𝑃𝐶))) |
4 | ofcfeqd2.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶)) | |
5 | 4 | ralrimiva 3125 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶)) |
6 | 5 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶)) |
7 | ofcfeqd2.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) | |
8 | 3, 6, 7 | rspcdva 3534 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑥)𝑃𝐶)) |
9 | 8 | mpteq2dva 5018 | . 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 2772 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
14 | 10, 11, 12, 13 | ofcfval 31033 | . 2 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
15 | 10, 11, 12, 13 | ofcfval 31033 | . 2 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑃𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑃𝐶))) |
16 | 9, 14, 15 | 3eqtr4d 2817 | 1 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝐹∘𝑓/𝑐𝑃𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3081 ↦ cmpt 5004 Fn wfn 6180 ‘cfv 6185 (class class class)co 6974 ∘𝑓/𝑐cofc 31030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-ofc 31031 |
This theorem is referenced by: coinfliplem 31414 |
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