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Theorem f1co 6741

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 6740	. 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–1-1→𝐶)
2		f1f 6730	. . . . . 6 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵)
3		fimacnv 6684	. . . . . 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 2742	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐴 = (^◡𝐺 “ 𝐵))
7		f1eq2 6726	. . 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 1541 ^◡ccnv 5623 “ cima 5627 ∘ ccom 5628 ⟶wf 6488 –1-1→wf1 6489

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-br 5099 df-opab 5161 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497

This theorem is referenced by: f1oco 6797 f1cofveqaeqALT 7204 tposf12 8193 domtr 8944 domtrfil 9116 dfac12lem2 10055 fin23lem28 10250 pwfseqlem5 10574 cofth 17861 injsubmefmnd 18822 gsumzf1o 19841 cycpmconjv 33224 erdsze2lem2 35398 fcoresf1b 47316 fundcmpsurinjpreimafv 47654

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