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Mirrors > Home > MPE Home > Th. List > offveqb | Structured version Visualization version GIF version |
Description: Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
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) |
Ref | Expression |
---|---|
offveqb | ⊢ (𝜑 → (𝐻 = (𝐹 ∘f 𝑅𝐺) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offveq.4 | . . . 4 ⊢ (𝜑 → 𝐻 Fn 𝐴) | |
2 | dffn5 6940 | . . . 4 ⊢ (𝐻 Fn 𝐴 ↔ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥))) | |
3 | 1, 2 | sylib 217 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥))) |
4 | offveq.2 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
5 | offveq.3 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐴) | |
6 | offveq.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | inidm 4210 | . . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
8 | offveq.5 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) | |
9 | offveq.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶) | |
10 | 4, 5, 6, 6, 7, 8, 9 | offval 7672 | . . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
11 | 3, 10 | eqeq12d 2740 | . 2 ⊢ (𝜑 → (𝐻 = (𝐹 ∘f 𝑅𝐺) ↔ (𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))) |
12 | fvexd 6896 | . . . 4 ⊢ (𝜑 → (𝐻‘𝑥) ∈ V) | |
13 | 12 | ralrimivw 3142 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) ∈ V) |
14 | mpteqb 7007 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐻‘𝑥) ∈ V → ((𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶))) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶))) |
16 | 11, 15 | bitrd 279 | 1 ⊢ (𝜑 → (𝐻 = (𝐹 ∘f 𝑅𝐺) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 ↦ cmpt 5221 Fn wfn 6528 ‘cfv 6533 (class class class)co 7401 ∘f cof 7661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 |
This theorem is referenced by: eqlkr2 38426 |
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