f1co - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > f1co		Structured version Visualization version GIF version

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 2740	. . 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 1536 ^◡ccnv 5687 “ cima 5691 ∘ ccom 5692 ⟶wf 6558 –1-1→wf1 6559

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567

This theorem is referenced by: f1oco 6871 f1cofveqaeqALT 7278 tposf12 8274 domtr 9045 domtrfil 9229 dfac12lem2 10182 fin23lem28 10377 pwfseqlem5 10700 cofth 17988 injsubmefmnd 18922 gsumzf1o 19944 cycpmconjv 33144 erdsze2lem2 35188 fcoresf1b 47019 fundcmpsurinjpreimafv 47332

W3C validator