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Theorem f1cof1 6820

Description: Composition of two one-to-one functions. Generalization of f1co 6821. (Contributed by AV, 18-Sep-2024.)

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

**Proof of Theorem f1cof1**
Step	Hyp	Ref	Expression
1		df-f1 6571	. . 3 ⊢ (𝐹:𝐶⟶1-1𝐷 ↔ (𝐹:𝐶⟶𝐷 ∧ Fun ^◡𝐹))
2		df-f1 6571	. . 3 ⊢ (𝐺:𝐴⟶1-1𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺))
3		ffun 6742	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → Fun 𝐺)
4		fcof 6763	. . . . . 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 6608	. . . . . . 7 ⊢ ((Fun ^◡𝐺 ∧ Fun ^◡𝐹) → Fun (^◡𝐺 ∘ ^◡𝐹))
7		cnvco 5879	. . . . . . . 8 ⊢ ^◡(𝐹 ∘ 𝐺) = (^◡𝐺 ∘ ^◡𝐹)
8	7	funeqi 6588	. . . . . . 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⟶1-1𝐷 ∧ 𝐺:𝐴⟶1-1𝐵) → ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
14		df-f1 6571	. 2 ⊢ ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶1-1𝐷 ↔ ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
15	13, 14	sylibr 237	1 ⊢ ((𝐹:𝐶⟶1-1𝐷 ∧ 𝐺:𝐴⟶1-1𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶1-1𝐷)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ^◡ccnv 5657 “ cima 5661 ∘ ccom 5662 Fun wfun 6560 ⟶wf 6562 ⟶1-1wf1 6563

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 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 2102 df-mo 2567 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4841 df-br 5109 df-opab 5174 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571

This theorem is used by: f1co 6821 f1cof1b 48420

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