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Mirrors > Home > ILE Home > Th. List > oprab2co | GIF version |
Description: Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.) |
Ref | Expression |
---|---|
oprab2co.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑅) |
oprab2co.2 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑆) |
oprab2co.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 〈𝐶, 𝐷〉) |
oprab2co.4 | ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑀𝐷)) |
Ref | Expression |
---|---|
oprab2co | ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oprab2co.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑅) | |
2 | oprab2co.2 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑆) | |
3 | opelxpi 4566 | . . 3 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆) → 〈𝐶, 𝐷〉 ∈ (𝑅 × 𝑆)) | |
4 | 1, 2, 3 | syl2anc 408 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 〈𝐶, 𝐷〉 ∈ (𝑅 × 𝑆)) |
5 | oprab2co.3 | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 〈𝐶, 𝐷〉) | |
6 | oprab2co.4 | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑀𝐷)) | |
7 | df-ov 5770 | . . . . 5 ⊢ (𝐶𝑀𝐷) = (𝑀‘〈𝐶, 𝐷〉) | |
8 | 7 | a1i 9 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐶𝑀𝐷) = (𝑀‘〈𝐶, 𝐷〉)) |
9 | 8 | mpoeq3ia 5829 | . . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑀𝐷)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑀‘〈𝐶, 𝐷〉)) |
10 | 6, 9 | eqtri 2158 | . 2 ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑀‘〈𝐶, 𝐷〉)) |
11 | 4, 5, 10 | oprabco 6107 | 1 ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 〈cop 3525 × cxp 4532 ∘ ccom 4538 Fn wfn 5113 ‘cfv 5118 (class class class)co 5767 ∈ cmpo 5769 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 |
This theorem is referenced by: (None) |
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