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Mirrors > Home > MPE Home > Th. List > fcoOLD | Structured version Visualization version GIF version |
Description: Obsolete version of fco 6732 as of 20-Sep-2024. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
fcoOLD | ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f 6538 | . . 3 ⊢ (𝐹:𝐵⟶𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶)) | |
2 | df-f 6538 | . . 3 ⊢ (𝐺:𝐴⟶𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵)) | |
3 | fnco 6658 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐴) | |
4 | 3 | 3expib 1119 | . . . . . 6 ⊢ (𝐹 Fn 𝐵 → ((𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐴)) |
5 | 4 | adantr 480 | . . . . 5 ⊢ ((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ((𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐴)) |
6 | rncoss 5962 | . . . . . . 7 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 | |
7 | sstr 3983 | . . . . . . 7 ⊢ ((ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ 𝐶) → ran (𝐹 ∘ 𝐺) ⊆ 𝐶) | |
8 | 6, 7 | mpan 687 | . . . . . 6 ⊢ (ran 𝐹 ⊆ 𝐶 → ran (𝐹 ∘ 𝐺) ⊆ 𝐶) |
9 | 8 | adantl 481 | . . . . 5 ⊢ ((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ran (𝐹 ∘ 𝐺) ⊆ 𝐶) |
10 | 5, 9 | jctird 526 | . . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ((𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶))) |
11 | 10 | imp 406 | . . 3 ⊢ (((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) ∧ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵)) → ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶)) |
12 | 1, 2, 11 | syl2anb 597 | . 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶)) |
13 | df-f 6538 | . 2 ⊢ ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ↔ ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶)) | |
14 | 12, 13 | sylibr 233 | 1 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ⊆ wss 3941 ran crn 5668 ∘ ccom 5671 Fn wfn 6529 ⟶wf 6530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-fun 6536 df-fn 6537 df-f 6538 |
This theorem is referenced by: (None) |
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