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Theorem f1co 6815

Description: Composition of one-to-one functions when the codomain of the first matches the domain of the second. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.) (Proof shortened by AV, 20-Sep-2024.)

**Assertion**
Ref	Expression
f1co	⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶)

**Proof of Theorem f1co**
Step	Hyp	Ref	Expression
1		f1cof1 6814	. 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–1-1→𝐶)
2		f1f 6804	. . . . . 6 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵)
3		fimacnv 6758	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → (^◡𝐺 “ 𝐵) = 𝐴)
4	2, 3	syl 17	. . . . 5 ⊢ (𝐺:𝐴–1-1→𝐵 → (^◡𝐺 “ 𝐵) = 𝐴)
5	4	adantl 481	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (^◡𝐺 “ 𝐵) = 𝐴)
6	5	eqcomd 2743	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐴 = (^◡𝐺 “ 𝐵))
7		f1eq2 6800	. . 3 ⊢ (𝐴 = (^◡𝐺 “ 𝐵) → ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ↔ (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–1-1→𝐶))
8	6, 7	syl 17	. 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ↔ (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–1-1→𝐶))
9	1, 8	mpbird 257	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 ↔ wb 206 ∧ wa 395 = wceq 1540 ^◡ccnv 5684 “ cima 5688 ∘ ccom 5689 ⟶wf 6557 –1-1→wf1 6558

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

This theorem is referenced by: f1oco 6871 f1cofveqaeqALT 7279 tposf12 8276 domtr 9047 domtrfil 9232 dfac12lem2 10185 fin23lem28 10380 pwfseqlem5 10703 cofth 17982 injsubmefmnd 18910 gsumzf1o 19930 cycpmconjv 33162 erdsze2lem2 35209 fcoresf1b 47082 fundcmpsurinjpreimafv 47395

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