f1cof1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > f1cof1		Structured version Visualization version GIF version

Theorem f1cof1 6772

Description: Composition of two one-to-one functions. Generalization of f1co 6773. (Contributed by AV, 18-Sep-2024.)

**Assertion**
Ref	Expression
f1cof1	⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)–1-1→𝐷)

**Proof of Theorem f1cof1**
Step	Hyp	Ref	Expression
1		df-f1 6526	. . 3 ⊢ (𝐹:𝐶–1-1→𝐷 ↔ (𝐹:𝐶⟶𝐷 ∧ Fun ^◡𝐹))
2		df-f1 6526	. . 3 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺))
3		ffun 6694	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → Fun 𝐺)
4		fcof 6715	. . . . . 6 ⊢ ((𝐹:𝐶⟶𝐷 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷)
5	3, 4	sylan2 602	. . . . 5 ⊢ ((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷)
6		funco 6561	. . . . . . 7 ⊢ ((Fun ^◡𝐺 ∧ Fun ^◡𝐹) → Fun (^◡𝐺 ∘ ^◡𝐹))
7		cnvco 5861	. . . . . . . 8 ⊢ ^◡(𝐹 ∘ 𝐺) = (^◡𝐺 ∘ ^◡𝐹)
8	7	funeqi 6542	. . . . . . 7 ⊢ (Fun ^◡(𝐹 ∘ 𝐺) ↔ Fun (^◡𝐺 ∘ ^◡𝐹))
9	6, 8	sylibr 236	. . . . . 6 ⊢ ((Fun ^◡𝐺 ∧ Fun ^◡𝐹) → Fun ^◡(𝐹 ∘ 𝐺))
10	9	ancoms 462	. . . . 5 ⊢ ((Fun ^◡𝐹 ∧ Fun ^◡𝐺) → Fun ^◡(𝐹 ∘ 𝐺))
11	5, 10	anim12i 622	. . . 4 ⊢ (((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) ∧ (Fun ^◡𝐹 ∧ Fun ^◡𝐺)) → ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
12	11	an4s 670	. . 3 ⊢ (((𝐹:𝐶⟶𝐷 ∧ Fun ^◡𝐹) ∧ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺)) → ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
13	1, 2, 12	syl2anb 607	. 2 ⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
14		df-f1 6526	. 2 ⊢ ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)–1-1→𝐷 ↔ ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
15	13, 14	sylibr 236	1 ⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)–1-1→𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ^◡ccnv 5646 “ cima 5650 ∘ ccom 5651 Fun wfun 6515 ⟶wf 6517 –1-1→wf1 6518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-pr 5390

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ral 3077 df-rex 3087 df-rab 3415 df-v 3456 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-sn 4583 df-pr 4585 df-op 4589 df-br 5101 df-opab 5163 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526

This theorem is referenced by: f1co 6773 f1cof1b 47671

W3C validator