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Mirrors > Home > ILE Home > Th. List > f1oco | 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 5218 | . . 3 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶)) | |
2 | df-f1o 5218 | . . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐵 ↔ (𝐺:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–onto→𝐵)) | |
3 | f1co 5428 | . . . . 5 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) | |
4 | foco 5443 | . . . . 5 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶) | |
5 | 3, 4 | anim12i 338 | . . . 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 588 | . . 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 291 | . 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 5218 | . 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–1-1-onto→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)) | |
9 | 7, 8 | sylibr 134 | 1 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝐺:𝐴–1-1-onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1-onto→𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∘ ccom 4626 –1-1→wf1 5208 –onto→wfo 5209 –1-1-onto→wf1o 5210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 |
This theorem is referenced by: isotr 5810 ener 6772 hashfacen 10787 nnf1o 11355 summodclem3 11359 fsumf1o 11369 prodmodclem3 11554 fprodf1o 11567 eulerthlemh 12201 |
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