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Mirrors > Home > MPE Home > Th. List > f1co | Structured version Visualization version GIF version |
Description: Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.) |
Ref | Expression |
---|---|
f1co | ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1 5893 | . . 3 ⊢ (𝐹:𝐵–1-1→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ Fun ◡𝐹)) | |
2 | df-f1 5893 | . . 3 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) | |
3 | fco 6058 | . . . . 5 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | |
4 | funco 5928 | . . . . . . 7 ⊢ ((Fun ◡𝐺 ∧ Fun ◡𝐹) → Fun (◡𝐺 ∘ ◡𝐹)) | |
5 | cnvco 5308 | . . . . . . . 8 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹) | |
6 | 5 | funeqi 5909 | . . . . . . 7 ⊢ (Fun ◡(𝐹 ∘ 𝐺) ↔ Fun (◡𝐺 ∘ ◡𝐹)) |
7 | 4, 6 | sylibr 224 | . . . . . 6 ⊢ ((Fun ◡𝐺 ∧ Fun ◡𝐹) → Fun ◡(𝐹 ∘ 𝐺)) |
8 | 7 | ancoms 469 | . . . . 5 ⊢ ((Fun ◡𝐹 ∧ Fun ◡𝐺) → Fun ◡(𝐹 ∘ 𝐺)) |
9 | 3, 8 | anim12i 590 | . . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) ∧ (Fun ◡𝐹 ∧ Fun ◡𝐺)) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
10 | 9 | an4s 869 | . . 3 ⊢ (((𝐹:𝐵⟶𝐶 ∧ Fun ◡𝐹) ∧ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
11 | 1, 2, 10 | syl2anb 496 | . 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
12 | df-f1 5893 | . 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ◡(𝐹 ∘ 𝐺))) | |
13 | 11, 12 | sylibr 224 | 1 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ◡ccnv 5113 ∘ ccom 5118 Fun wfun 5882 ⟶wf 5884 –1-1→wf1 5885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 |
This theorem is referenced by: f1oco 6159 f1cofveqaeqALT 6516 tposf12 7377 domtr 8009 dfac12lem2 8966 fin23lem28 9162 pwfseqlem5 9485 cofth 16595 gsumzf1o 18313 erdsze2lem2 31186 |
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