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Mirrors > Home > ILE Home > Th. List > fmptcos | GIF version |
Description: Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fmptcof.1 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵) |
fmptcof.2 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅)) |
fmptcof.3 | ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆)) |
Ref | Expression |
---|---|
fmptcos | ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptcof.1 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵) | |
2 | fmptcof.2 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅)) | |
3 | fmptcof.3 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆)) | |
4 | nfcv 2281 | . . . 4 ⊢ Ⅎ𝑧𝑆 | |
5 | nfcsb1v 3035 | . . . 4 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝑆 | |
6 | csbeq1a 3012 | . . . 4 ⊢ (𝑦 = 𝑧 → 𝑆 = ⦋𝑧 / 𝑦⦌𝑆) | |
7 | 4, 5, 6 | cbvmpt 4023 | . . 3 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑧 ∈ 𝐵 ↦ ⦋𝑧 / 𝑦⦌𝑆) |
8 | 3, 7 | syl6eq 2188 | . 2 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐵 ↦ ⦋𝑧 / 𝑦⦌𝑆)) |
9 | csbeq1 3006 | . 2 ⊢ (𝑧 = 𝑅 → ⦋𝑧 / 𝑦⦌𝑆 = ⦋𝑅 / 𝑦⦌𝑆) | |
10 | 1, 2, 8, 9 | fmptcof 5587 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ⦋csb 3003 ↦ cmpt 3989 ∘ ccom 4543 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 |
This theorem is referenced by: fmpoco 6113 |
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