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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 2255 | . . . 4 ⊢ Ⅎ𝑧𝑆 | |
5 | nfcsb1v 3001 | . . . 4 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝑆 | |
6 | csbeq1a 2979 | . . . 4 ⊢ (𝑦 = 𝑧 → 𝑆 = ⦋𝑧 / 𝑦⦌𝑆) | |
7 | 4, 5, 6 | cbvmpt 3983 | . . 3 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑧 ∈ 𝐵 ↦ ⦋𝑧 / 𝑦⦌𝑆) |
8 | 3, 7 | syl6eq 2163 | . 2 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐵 ↦ ⦋𝑧 / 𝑦⦌𝑆)) |
9 | csbeq1 2974 | . 2 ⊢ (𝑧 = 𝑅 → ⦋𝑧 / 𝑦⦌𝑆 = ⦋𝑅 / 𝑦⦌𝑆) | |
10 | 1, 2, 8, 9 | fmptcof 5541 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1314 ∈ wcel 1463 ∀wral 2390 ⦋csb 2971 ↦ cmpt 3949 ∘ ccom 4503 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-fv 5089 |
This theorem is referenced by: fmpoco 6067 |
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