Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcomptf | Structured version Visualization version GIF version |
Description: Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 6887. (Contributed by Thierry Arnoux, 30-Jun-2017.) |
Ref | Expression |
---|---|
fcomptf.1 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
fcomptf | ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
3 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑥𝐸 | |
4 | 1, 2, 3 | nff 6503 | . . . 4 ⊢ Ⅎ𝑥 𝐴:𝐷⟶𝐸 |
5 | fcomptf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
7 | 5, 6, 2 | nff 6503 | . . . 4 ⊢ Ⅎ𝑥 𝐵:𝐶⟶𝐷 |
8 | 4, 7 | nfan 1891 | . . 3 ⊢ Ⅎ𝑥(𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) |
9 | ffvelrn 6841 | . . . . 5 ⊢ ((𝐵:𝐶⟶𝐷 ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷) | |
10 | 9 | adantll 710 | . . . 4 ⊢ (((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷) |
11 | 10 | ex 413 | . . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝑥 ∈ 𝐶 → (𝐵‘𝑥) ∈ 𝐷)) |
12 | 8, 11 | ralrimi 3213 | . 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → ∀𝑥 ∈ 𝐶 (𝐵‘𝑥) ∈ 𝐷) |
13 | ffn 6507 | . . . 4 ⊢ (𝐵:𝐶⟶𝐷 → 𝐵 Fn 𝐶) | |
14 | 13 | adantl 482 | . . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 Fn 𝐶) |
15 | 5 | dffn5f 6729 | . . 3 ⊢ (𝐵 Fn 𝐶 ↔ 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥))) |
16 | 14, 15 | sylib 219 | . 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥))) |
17 | ffn 6507 | . . . 4 ⊢ (𝐴:𝐷⟶𝐸 → 𝐴 Fn 𝐷) | |
18 | 17 | adantr 481 | . . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 Fn 𝐷) |
19 | dffn5 6717 | . . 3 ⊢ (𝐴 Fn 𝐷 ↔ 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦))) | |
20 | 18, 19 | sylib 219 | . 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦))) |
21 | fveq2 6663 | . 2 ⊢ (𝑦 = (𝐵‘𝑥) → (𝐴‘𝑦) = (𝐴‘(𝐵‘𝑥))) | |
22 | 12, 16, 20, 21 | fmptcof 6884 | 1 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Ⅎwnfc 2958 ↦ cmpt 5137 ∘ ccom 5552 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 |
This theorem is referenced by: ofoprabco 30337 |
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