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Mirrors > Home > MPE Home > Th. List > foco | Structured version Visualization version GIF version |
Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.) (Proof shortened by AV, 29-Sep-2024.) |
Ref | Expression |
---|---|
foco | ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → 𝐹:𝐵–onto→𝐶) | |
2 | fofun 6822 | . . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → Fun 𝐺) | |
3 | 2 | adantl 481 | . . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → Fun 𝐺) |
4 | forn 6824 | . . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → ran 𝐺 = 𝐵) | |
5 | eqimss2 4055 | . . . . 5 ⊢ (ran 𝐺 = 𝐵 → 𝐵 ⊆ ran 𝐺) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐵 ⊆ ran 𝐺) |
7 | 6 | adantl 481 | . . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → 𝐵 ⊆ ran 𝐺) |
8 | focofo 6834 | . . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ Fun 𝐺 ∧ 𝐵 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐵)–onto→𝐶) | |
9 | 1, 3, 7, 8 | syl3anc 1370 | . 2 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐵)–onto→𝐶) |
10 | focnvimacdmdm 6833 | . . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = 𝐴) | |
11 | 10 | eqcomd 2741 | . . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐴 = (◡𝐺 “ 𝐵)) |
12 | 11 | adantl 481 | . . 3 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → 𝐴 = (◡𝐺 “ 𝐵)) |
13 | foeq2 6818 | . . 3 ⊢ (𝐴 = (◡𝐺 “ 𝐵) → ((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ↔ (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐵)–onto→𝐶)) | |
14 | 12, 13 | syl 17 | . 2 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → ((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ↔ (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐵)–onto→𝐶)) |
15 | 9, 14 | mpbird 257 | 1 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ⊆ wss 3963 ◡ccnv 5688 ran crn 5690 “ cima 5692 ∘ ccom 5693 Fun wfun 6557 –onto→wfo 6561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 |
This theorem is referenced by: f1oco 6872 wdomtr 9613 fin1a2lem7 10444 cofull 17988 sursubmefmnd 18922 uniiccdif 25627 fcoresfob 47022 |
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