|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 6966 | . . . 4 ⊢ (𝐻 Fn 𝐴 ↔ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥))) | |
| 3 | 1, 2 | sylib 218 | . . 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 4226 | . . . 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 7707 | . . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) | 
| 11 | 3, 10 | eqeq12d 2752 | . 2 ⊢ (𝜑 → (𝐻 = (𝐹 ∘f 𝑅𝐺) ↔ (𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))) | 
| 12 | fvexd 6920 | . . . 4 ⊢ (𝜑 → (𝐻‘𝑥) ∈ V) | |
| 13 | 12 | ralrimivw 3149 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) ∈ V) | 
| 14 | mpteqb 7034 | . . 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 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ↦ cmpt 5224 Fn wfn 6555 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 | 
| This theorem is referenced by: eqlkr2 39102 | 
| Copyright terms: Public domain | W3C validator |