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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.) |
Ref | Expression |
---|---|
foco | ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffo2 6569 | . . 3 ⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶)) | |
2 | dffo2 6569 | . . 3 ⊢ (𝐺:𝐴–onto→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) | |
3 | fco 6505 | . . . . 5 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | |
4 | 3 | ad2ant2r 746 | . . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
5 | fdm 6495 | . . . . . . . 8 ⊢ (𝐹:𝐵⟶𝐶 → dom 𝐹 = 𝐵) | |
6 | eqtr3 2820 | . . . . . . . 8 ⊢ ((dom 𝐹 = 𝐵 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺) | |
7 | 5, 6 | sylan 583 | . . . . . . 7 ⊢ ((𝐹:𝐵⟶𝐶 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺) |
8 | rncoeq 5811 | . . . . . . . . 9 ⊢ (dom 𝐹 = ran 𝐺 → ran (𝐹 ∘ 𝐺) = ran 𝐹) | |
9 | 8 | eqeq1d 2800 | . . . . . . . 8 ⊢ (dom 𝐹 = ran 𝐺 → (ran (𝐹 ∘ 𝐺) = 𝐶 ↔ ran 𝐹 = 𝐶)) |
10 | 9 | biimpar 481 | . . . . . . 7 ⊢ ((dom 𝐹 = ran 𝐺 ∧ ran 𝐹 = 𝐶) → ran (𝐹 ∘ 𝐺) = 𝐶) |
11 | 7, 10 | sylan 583 | . . . . . 6 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐺 = 𝐵) ∧ ran 𝐹 = 𝐶) → ran (𝐹 ∘ 𝐺) = 𝐶) |
12 | 11 | an32s 651 | . . . . 5 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ ran 𝐺 = 𝐵) → ran (𝐹 ∘ 𝐺) = 𝐶) |
13 | 12 | adantrl 715 | . . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) → ran (𝐹 ∘ 𝐺) = 𝐶) |
14 | 4, 13 | jca 515 | . . 3 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶)) |
15 | 1, 2, 14 | syl2anb 600 | . 2 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶)) |
16 | dffo2 6569 | . 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶)) | |
17 | 15, 16 | sylibr 237 | 1 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 dom cdm 5519 ran crn 5520 ∘ ccom 5523 ⟶wf 6320 –onto→wfo 6322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 |
This theorem is referenced by: f1oco 6612 wdomtr 9023 fin1a2lem7 9817 cofull 17196 sursubmefmnd 18053 uniiccdif 24182 |
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