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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 5162 | . . . 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 3140 | . . . 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 5661 | . . 3 ⊢ (φ → (𝐹 ∘𝑓 𝑅𝐺) = (x ∈ A ↦ (B𝑅𝐶))) |
11 | 3, 10 | eqeq12d 2051 | . 2 ⊢ (φ → (𝐻 = (𝐹 ∘𝑓 𝑅𝐺) ↔ (x ∈ A ↦ (𝐻‘x)) = (x ∈ A ↦ (B𝑅𝐶)))) |
12 | funfvex 5135 | . . . . . 6 ⊢ ((Fun 𝐻 ∧ x ∈ dom 𝐻) → (𝐻‘x) ∈ V) | |
13 | 12 | funfni 4942 | . . . . 5 ⊢ ((𝐻 Fn A ∧ x ∈ A) → (𝐻‘x) ∈ V) |
14 | 1, 13 | sylan 267 | . . . 4 ⊢ ((φ ∧ x ∈ A) → (𝐻‘x) ∈ V) |
15 | 14 | ralrimiva 2386 | . . 3 ⊢ (φ → ∀x ∈ A (𝐻‘x) ∈ V) |
16 | mpteqb 5204 | . . 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 177 | 1 ⊢ (φ → (𝐻 = (𝐹 ∘𝑓 𝑅𝐺) ↔ ∀x ∈ A (𝐻‘x) = (B𝑅𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ∀wral 2300 Vcvv 2551 ↦ cmpt 3809 Fn wfn 4840 ‘cfv 4845 (class class class)co 5455 ∘𝑓 cof 5652 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-setind 4220 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-of 5654 |
This theorem is referenced by: (None) |
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