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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 6365 | . . 3 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶)) | |
2 | df-f1o 6365 | . . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐵 ↔ (𝐺:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–onto→𝐵)) | |
3 | f1co 6605 | . . . . 5 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) | |
4 | foco 6625 | . . . . 5 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶) | |
5 | 3, 4 | anim12i 616 | . . . 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 601 | . 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 6365 | . 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–1-1-onto→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)) | |
9 | 7, 8 | sylibr 237 | 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 399 ∘ ccom 5540 –1-1→wf1 6355 –onto→wfo 6356 –1-1-onto→wf1o 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 |
This theorem is referenced by: fveqf1o 7091 f1ofvswap 7094 isotr 7123 ener 8653 omf1o 8726 enfixsn 8732 entrfil 8840 oef1o 9291 cnfcom3 9297 infxpenc 9597 ackbij2lem2 9819 canthp1lem2 10232 pwfseqlem5 10242 hashfacen 13983 hashfacenOLD 13984 summolem3 15243 fsumf1o 15252 ackbijnn 15355 prodmolem3 15458 fprodf1o 15471 eulerthlem2 16298 symgcl 18731 pmtrfconj 18812 gsumval3eu 19243 gsumval3lem1 19244 gsumval3 19246 lmimco 20760 resinf1o 25379 motco 26585 counop 29956 symgcom 31025 pmtrcnel 31031 cycpmcl 31056 cycpmconjslem2 31095 cycpmconjs 31096 eulerpartgbij 32005 derangenlem 32800 subfacp1lem5 32813 poimirlem9 35472 poimirlem15 35478 poimirlem16 35479 poimirlem17 35480 poimirlem19 35482 poimirlem20 35483 rngoisoco 35826 lautco 37797 metakunt34 39821 clsneif1o 41332 neicvgf1o 41342 isomushgr 44894 isomgrtr 44907 uspgrbisymrelALT 44933 |
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