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Theorem f1co 6796

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 6795	. 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐵)–1-1→𝐶)
2		f1f 6784	. . . . . 6 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵)
3		fimacnv 6736	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → (^◡𝐺 “ 𝐵) = 𝐴)
4	2, 3	syl 17	. . . . 5 ⊢ (𝐺:𝐴–1-1→𝐵 → (^◡𝐺 “ 𝐵) = 𝐴)
5	4	adantl 483	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (^◡𝐺 “ 𝐵) = 𝐴)
6	5	eqcomd 2739	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐴 = (^◡𝐺 “ 𝐵))
7		f1eq2 6780	. . 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 205 ∧ wa 397 = wceq 1542 ^◡ccnv 5674 “ cima 5678 ∘ ccom 5679 ⟶wf 6536 –1-1→wf1 6537

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545

This theorem is referenced by: f1oco 6853 f1cofveqaeqALT 7253 tposf12 8231 domtr 8999 domtrfil 9191 dfac12lem2 10135 fin23lem28 10331 pwfseqlem5 10654 cofth 17882 injsubmefmnd 18774 gsumzf1o 19772 cycpmconjv 32279 erdsze2lem2 34133 fcoresf1b 45715 fundcmpsurinjpreimafv 46011

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