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Theorem f1cof1 6814

Description: Composition of two one-to-one functions. Generalization of f1co 6815. (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 6566	. . 3 ⊢ (𝐹:𝐶–1-1→𝐷 ↔ (𝐹:𝐶⟶𝐷 ∧ Fun ^◡𝐹))
2		df-f1 6566	. . 3 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺))
3		ffun 6739	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → Fun 𝐺)
4		fcof 6759	. . . . . 6 ⊢ ((𝐹:𝐶⟶𝐷 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷)
5	3, 4	sylan2 593	. . . . 5 ⊢ ((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷)
6		funco 6606	. . . . . . 7 ⊢ ((Fun ^◡𝐺 ∧ Fun ^◡𝐹) → Fun (^◡𝐺 ∘ ^◡𝐹))
7		cnvco 5896	. . . . . . . 8 ⊢ ^◡(𝐹 ∘ 𝐺) = (^◡𝐺 ∘ ^◡𝐹)
8	7	funeqi 6587	. . . . . . 7 ⊢ (Fun ^◡(𝐹 ∘ 𝐺) ↔ Fun (^◡𝐺 ∘ ^◡𝐹))
9	6, 8	sylibr 234	. . . . . 6 ⊢ ((Fun ^◡𝐺 ∧ Fun ^◡𝐹) → Fun ^◡(𝐹 ∘ 𝐺))
10	9	ancoms 458	. . . . 5 ⊢ ((Fun ^◡𝐹 ∧ Fun ^◡𝐺) → Fun ^◡(𝐹 ∘ 𝐺))
11	5, 10	anim12i 613	. . . 4 ⊢ (((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) ∧ (Fun ^◡𝐹 ∧ Fun ^◡𝐺)) → ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
12	11	an4s 660	. . 3 ⊢ (((𝐹:𝐶⟶𝐷 ∧ Fun ^◡𝐹) ∧ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺)) → ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
13	1, 2, 12	syl2anb 598	. 2 ⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
14		df-f1 6566	. 2 ⊢ ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)–1-1→𝐷 ↔ ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
15	13, 14	sylibr 234	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 395 ^◡ccnv 5684 “ cima 5688 ∘ ccom 5689 Fun wfun 6555 ⟶wf 6557 –1-1→wf1 6558

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566

This theorem is referenced by: f1co 6815 f1cof1b 47089

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