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Theorem f1cof1 6784

Description: Composition of two one-to-one functions. Generalization of f1co 6785. (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 6538	. . 3 ⊢ (𝐹:𝐶–1-1→𝐷 ↔ (𝐹:𝐶⟶𝐷 ∧ Fun ^◡𝐹))
2		df-f1 6538	. . 3 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺))
3		ffun 6706	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → Fun 𝐺)
4		fcof 6727	. . . . . 6 ⊢ ((𝐹:𝐶⟶𝐷 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷)
5	3, 4	sylan2 605	. . . . 5 ⊢ ((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷)
6		funco 6574	. . . . . . 7 ⊢ ((Fun ^◡𝐺 ∧ Fun ^◡𝐹) → Fun (^◡𝐺 ∘ ^◡𝐹))
7		cnvco 5869	. . . . . . . 8 ⊢ ^◡(𝐹 ∘ 𝐺) = (^◡𝐺 ∘ ^◡𝐹)
8	7	funeqi 6554	. . . . . . 7 ⊢ (Fun ^◡(𝐹 ∘ 𝐺) ↔ Fun (^◡𝐺 ∘ ^◡𝐹))
9	6, 8	sylibr 237	. . . . . 6 ⊢ ((Fun ^◡𝐺 ∧ Fun ^◡𝐹) → Fun ^◡(𝐹 ∘ 𝐺))
10	9	ancoms 464	. . . . 5 ⊢ ((Fun ^◡𝐹 ∧ Fun ^◡𝐺) → Fun ^◡(𝐹 ∘ 𝐺))
11	5, 10	anim12i 625	. . . 4 ⊢ (((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) ∧ (Fun ^◡𝐹 ∧ Fun ^◡𝐺)) → ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
12	11	an4s 673	. . 3 ⊢ (((𝐹:𝐶⟶𝐷 ∧ Fun ^◡𝐹) ∧ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺)) → ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
13	1, 2, 12	syl2anb 610	. 2 ⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
14		df-f1 6538	. 2 ⊢ ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)–1-1→𝐷 ↔ ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
15	13, 14	sylibr 237	1 ⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)–1-1→𝐷)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ^◡ccnv 5654 “ cima 5658 ∘ ccom 5659 Fun wfun 6527 ⟶wf 6529 –1-1→wf1 6530

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-pr 5398

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5168 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538

This theorem is used by: f1co 6785 f1cof1b 47966

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