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Mirrors > Home > ILE Home > Th. List > offveqb | 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 | dffn5im 5475 | . . . 4 ⊢ (𝐻 Fn 𝐴 → 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥))) | |
3 | 1, 2 | syl 14 | . . 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 3290 | . . . 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 5997 | . . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
11 | 3, 10 | eqeq12d 2155 | . 2 ⊢ (𝜑 → (𝐻 = (𝐹 ∘𝑓 𝑅𝐺) ↔ (𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))) |
12 | funfvex 5446 | . . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ dom 𝐻) → (𝐻‘𝑥) ∈ V) | |
13 | 12 | funfni 5231 | . . . . 5 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ V) |
14 | 1, 13 | sylan 281 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ V) |
15 | 14 | ralrimiva 2508 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) ∈ V) |
16 | mpteqb 5519 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐻‘𝑥) ∈ V → ((𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶))) | |
17 | 15, 16 | syl 14 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶))) |
18 | 11, 17 | bitrd 187 | 1 ⊢ (𝜑 → (𝐻 = (𝐹 ∘𝑓 𝑅𝐺) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 ∀wral 2417 Vcvv 2689 ↦ cmpt 3997 Fn wfn 5126 ‘cfv 5131 (class class class)co 5782 ∘𝑓 cof 5988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-of 5990 |
This theorem is referenced by: (None) |
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