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Mirrors > Home > ILE Home > Th. List > oprabco | GIF version |
Description: Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.) |
Ref | Expression |
---|---|
oprabco.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷) |
oprabco.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
oprabco.3 | ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶)) |
Ref | Expression |
---|---|
oprabco | ⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oprabco.1 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷) | |
2 | 1 | adantl 275 | . . 3 ⊢ ((𝐻 Fn 𝐷 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷) |
3 | oprabco.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
4 | 3 | a1i 9 | . . 3 ⊢ (𝐻 Fn 𝐷 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)) |
5 | dffn5im 5435 | . . 3 ⊢ (𝐻 Fn 𝐷 → 𝐻 = (𝑧 ∈ 𝐷 ↦ (𝐻‘𝑧))) | |
6 | fveq2 5389 | . . 3 ⊢ (𝑧 = 𝐶 → (𝐻‘𝑧) = (𝐻‘𝐶)) | |
7 | 2, 4, 5, 6 | fmpoco 6081 | . 2 ⊢ (𝐻 Fn 𝐷 → (𝐻 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶))) |
8 | oprabco.3 | . 2 ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶)) | |
9 | 7, 8 | syl6reqr 2169 | 1 ⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 ∘ ccom 4513 Fn wfn 5088 ‘cfv 5093 ∈ cmpo 5744 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 |
This theorem is referenced by: oprab2co 6083 |
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