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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 6568 | . . 3 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶)) | |
| 2 | df-f1o 6568 | . . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐵 ↔ (𝐺:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–onto→𝐵)) | |
| 3 | f1co 6815 | . . . . 5 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) | |
| 4 | foco 6834 | . . . . 5 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶) | |
| 5 | 3, 4 | anim12i 613 | . . . 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 660 | . . 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 598 | . 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 6568 | . 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–1-1-onto→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)) | |
| 9 | 7, 8 | sylibr 234 | 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 395 ∘ ccom 5689 –1-1→wf1 6558 –onto→wfo 6559 –1-1-onto→wf1o 6560 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 |
| This theorem is referenced by: fveqf1o 7322 f1ocoima 7323 f1ofvswap 7326 isotr 7356 ener 9041 omf1o 9115 enfixsn 9121 entrfil 9225 oef1o 9738 cnfcom3 9744 infxpenc 10058 ackbij2lem2 10279 canthp1lem2 10693 pwfseqlem5 10703 hashfacen 14493 summolem3 15750 fsumf1o 15759 ackbijnn 15864 prodmolem3 15969 fprodf1o 15982 eulerthlem2 16819 symgcl 19402 pmtrfconj 19484 gsumval3eu 19922 gsumval3lem1 19923 gsumval3 19925 lmimco 21864 resinf1o 26578 motco 28548 counop 31940 symgcom 33103 pmtrcnel 33109 cycpmcl 33136 cycpmconjslem2 33175 cycpmconjs 33176 1arithidomlem2 33564 eulerpartgbij 34374 derangenlem 35176 subfacp1lem5 35189 poimirlem9 37636 poimirlem15 37642 poimirlem16 37643 poimirlem17 37644 poimirlem19 37646 poimirlem20 37647 rngoisoco 37989 lautco 40099 metakunt34 42239 clsneif1o 44117 neicvgf1o 44127 grimco 47880 gricushgr 47886 grlictr 47975 uspgrbisymrelALT 48071 |
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