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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 7080. (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 2904 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
3 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥𝐸 | |
4 | 1, 2, 3 | nff 6665 | . . . 4 ⊢ Ⅎ𝑥 𝐴:𝐷⟶𝐸 |
5 | fcomptf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
7 | 5, 6, 2 | nff 6665 | . . . 4 ⊢ Ⅎ𝑥 𝐵:𝐶⟶𝐷 |
8 | 4, 7 | nfan 1903 | . . 3 ⊢ Ⅎ𝑥(𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) |
9 | ffvelcdm 7033 | . . . . 5 ⊢ ((𝐵:𝐶⟶𝐷 ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷) | |
10 | 9 | adantll 713 | . . . 4 ⊢ (((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) ∧ 𝑥 ∈ 𝐶) → (𝐵‘𝑥) ∈ 𝐷) |
11 | 10 | ex 414 | . . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝑥 ∈ 𝐶 → (𝐵‘𝑥) ∈ 𝐷)) |
12 | 8, 11 | ralrimi 3239 | . 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → ∀𝑥 ∈ 𝐶 (𝐵‘𝑥) ∈ 𝐷) |
13 | ffn 6669 | . . . 4 ⊢ (𝐵:𝐶⟶𝐷 → 𝐵 Fn 𝐶) | |
14 | 13 | adantl 483 | . . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 Fn 𝐶) |
15 | 5 | dffn5f 6914 | . . 3 ⊢ (𝐵 Fn 𝐶 ↔ 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥))) |
16 | 14, 15 | sylib 217 | . 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐵 = (𝑥 ∈ 𝐶 ↦ (𝐵‘𝑥))) |
17 | ffn 6669 | . . . 4 ⊢ (𝐴:𝐷⟶𝐸 → 𝐴 Fn 𝐷) | |
18 | 17 | adantr 482 | . . 3 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 Fn 𝐷) |
19 | dffn5 6902 | . . 3 ⊢ (𝐴 Fn 𝐷 ↔ 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦))) | |
20 | 18, 19 | sylib 217 | . 2 ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → 𝐴 = (𝑦 ∈ 𝐷 ↦ (𝐴‘𝑦))) |
21 | fveq2 6843 | . 2 ⊢ (𝑦 = (𝐵‘𝑥) → (𝐴‘𝑦) = (𝐴‘(𝐵‘𝑥))) | |
22 | 12, 16, 20, 21 | fmptcof 7077 | 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 397 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2884 ↦ cmpt 5189 ∘ ccom 5638 Fn wfn 6492 ⟶wf 6493 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 |
This theorem is referenced by: ofoprabco 31626 |
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