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 6895. (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 2977 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
3 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥𝐸 | |
4 | 1, 2, 3 | nff 6510 | . . . 4 ⊢ Ⅎ𝑥 𝐴:𝐷⟶𝐸 |
5 | fcomptf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
7 | 5, 6, 2 | nff 6510 | . . . 4 ⊢ Ⅎ𝑥 𝐵:𝐶⟶𝐷 |
8 | 4, 7 | nfan 1900 | . . 3 ⊢ Ⅎ𝑥(𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) |
9 | ffvelrn 6849 | . . . . 5 ⊢ ((𝐵:𝐶⟶𝐷 ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷) | |
10 | 9 | adantll 712 | . . . 4 ⊢ (((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷) |
11 | 10 | ex 415 | . . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝑥 ∈ 𝐶 → (𝐵‘𝑥) ∈ 𝐷)) |
12 | 8, 11 | ralrimi 3216 | . 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → ∀𝑥 ∈ 𝐶 (𝐵‘𝑥) ∈ 𝐷) |
13 | ffn 6514 | . . . 4 ⊢ (𝐵:𝐶⟶𝐷 → 𝐵 Fn 𝐶) | |
14 | 13 | adantl 484 | . . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 Fn 𝐶) |
15 | 5 | dffn5f 6736 | . . 3 ⊢ (𝐵 Fn 𝐶 ↔ 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥))) |
16 | 14, 15 | sylib 220 | . 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥))) |
17 | ffn 6514 | . . . 4 ⊢ (𝐴:𝐷⟶𝐸 → 𝐴 Fn 𝐷) | |
18 | 17 | adantr 483 | . . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 Fn 𝐷) |
19 | dffn5 6724 | . . 3 ⊢ (𝐴 Fn 𝐷 ↔ 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦))) | |
20 | 18, 19 | sylib 220 | . 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦))) |
21 | fveq2 6670 | . 2 ⊢ (𝑦 = (𝐵‘𝑥) → (𝐴‘𝑦) = (𝐴‘(𝐵‘𝑥))) | |
22 | 12, 16, 20, 21 | fmptcof 6892 | 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 398 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2961 ↦ cmpt 5146 ∘ ccom 5559 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 |
This theorem is referenced by: ofoprabco 30409 |
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