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Mirrors > Home > MPE Home > Th. List > f1oco | Structured version Visualization version GIF version |
Description: Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.) |
Ref | Expression |
---|---|
f1oco | ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝐺:𝐴–1-1-onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1-onto→𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1o 6357 | . . 3 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶)) | |
2 | df-f1o 6357 | . . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐵 ↔ (𝐺:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–onto→𝐵)) | |
3 | f1co 6580 | . . . . 5 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) | |
4 | foco 6597 | . . . . 5 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶) | |
5 | 3, 4 | anim12i 614 | . . . 4 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵)) → ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)) |
6 | 5 | an4s 658 | . . 3 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶) ∧ (𝐺:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–onto→𝐵)) → ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)) |
7 | 1, 2, 6 | syl2anb 599 | . 2 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝐺:𝐴–1-1-onto→𝐵) → ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)) |
8 | df-f1o 6357 | . 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–1-1-onto→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)) | |
9 | 7, 8 | sylibr 236 | 1 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝐺:𝐴–1-1-onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1-onto→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∘ ccom 5554 –1-1→wf1 6347 –onto→wfo 6348 –1-1-onto→wf1o 6349 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 |
This theorem is referenced by: fveqf1o 7052 isotr 7083 ener 8550 omf1o 8614 enfixsn 8620 oef1o 9155 cnfcom3 9161 infxpenc 9438 ackbij2lem2 9656 canthp1lem2 10069 pwfseqlem5 10079 hashfacen 13806 summolem3 15065 fsumf1o 15074 ackbijnn 15177 prodmolem3 15281 fprodf1o 15294 eulerthlem2 16113 symgcl 18507 pmtrfconj 18588 gsumval3eu 19018 gsumval3lem1 19019 gsumval3 19021 lmimco 20982 resinf1o 25114 motco 26320 counop 29692 symgcom 30722 pmtrcnel 30728 cycpmcl 30753 cycpmconjslem2 30792 cycpmconjs 30793 eulerpartgbij 31625 derangenlem 32413 subfacp1lem5 32426 poimirlem9 34895 poimirlem15 34901 poimirlem16 34902 poimirlem17 34903 poimirlem19 34905 poimirlem20 34906 rngoisoco 35254 lautco 37227 clsneif1o 40447 neicvgf1o 40457 isomushgr 43984 isomgrtr 43997 uspgrbisymrelALT 44023 |
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