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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 6500 | . . 3 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶)) | |
2 | df-f1o 6500 | . . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐵 ↔ (𝐺:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–onto→𝐵)) | |
3 | f1co 6747 | . . . . 5 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) | |
4 | foco 6767 | . . . . 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 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 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 6500 | . 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–1-1-onto→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)) | |
9 | 7, 8 | sylibr 233 | 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 396 ∘ ccom 5635 –1-1→wf1 6490 –onto→wfo 6491 –1-1-onto→wf1o 6492 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 |
This theorem is referenced by: fveqf1o 7245 f1ofvswap 7248 isotr 7277 ener 8937 omf1o 9015 enfixsn 9021 entrfil 9128 oef1o 9630 cnfcom3 9636 infxpenc 9950 ackbij2lem2 10172 canthp1lem2 10585 pwfseqlem5 10595 hashfacen 14343 hashfacenOLD 14344 summolem3 15591 fsumf1o 15600 ackbijnn 15705 prodmolem3 15808 fprodf1o 15821 eulerthlem2 16646 symgcl 19157 pmtrfconj 19239 gsumval3eu 19672 gsumval3lem1 19673 gsumval3 19675 lmimco 21235 resinf1o 25876 motco 27368 counop 30749 symgcom 31817 pmtrcnel 31823 cycpmcl 31848 cycpmconjslem2 31887 cycpmconjs 31888 eulerpartgbij 32841 derangenlem 33634 subfacp1lem5 33647 poimirlem9 36054 poimirlem15 36060 poimirlem16 36061 poimirlem17 36062 poimirlem19 36064 poimirlem20 36065 rngoisoco 36408 lautco 38527 metakunt34 40577 clsneif1o 42318 neicvgf1o 42328 isomushgr 45950 isomgrtr 45963 uspgrbisymrelALT 45989 |
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