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 | ofcfeqd2.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) | |
2 | ofcfeqd2.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶)) | |
3 | 2 | ralrimiva 2966 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶)) |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶)) |
5 | oveq1 6657 | . . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦𝑅𝐶) = ((𝐹‘𝑥)𝑅𝐶)) | |
6 | oveq1 6657 | . . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦𝑃𝐶) = ((𝐹‘𝑥)𝑃𝐶)) | |
7 | 5, 6 | eqeq12d 2637 | . . . . 5 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝑦𝑅𝐶) = (𝑦𝑃𝐶) ↔ ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑥)𝑃𝐶))) |
8 | 7 | rspcva 3307 | . . . 4 ⊢ (((𝐹‘𝑥) ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶)) → ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑥)𝑃𝐶)) |
9 | 1, 4, 8 | syl2anc 693 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑥)𝑃𝐶)) |
10 | 9 | mpteq2dva 4744 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑃𝐶))) |
11 | ofcfeqd2.3 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
12 | ofcfeqd2.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
13 | ofcfeqd2.5 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
14 | eqidd 2623 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
15 | 11, 12, 13, 14 | ofcfval 30160 | . 2 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
16 | 11, 12, 13, 14 | ofcfval 30160 | . 2 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑃𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑃𝐶))) |
17 | 10, 15, 16 | 3eqtr4d 2666 | 1 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝐹∘𝑓/𝑐𝑃𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ↦ cmpt 4729 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ∘𝑓/𝑐cofc 30157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-ofc 30158 |
This theorem is referenced by: coinfliplem 30540 |
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