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Theorem f1cof1 6786

Description: Composition of two one-to-one functions. Generalization of f1co 6787. (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 6541	. . 3 ⊢ (𝐹:𝐶–1-1→𝐷 ↔ (𝐹:𝐶⟶𝐷 ∧ Fun ^◡𝐹))
2		df-f1 6541	. . 3 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺))
3		ffun 6708	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → Fun 𝐺)
4		fcof 6729	. . . . . 6 ⊢ ((𝐹:𝐶⟶𝐷 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷)
5	3, 4	sylan2 604	. . . . 5 ⊢ ((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷)
6		funco 6576	. . . . . . 7 ⊢ ((Fun ^◡𝐺 ∧ Fun ^◡𝐹) → Fun (^◡𝐺 ∘ ^◡𝐹))
7		cnvco 5875	. . . . . . . 8 ⊢ ^◡(𝐹 ∘ 𝐺) = (^◡𝐺 ∘ ^◡𝐹)
8	7	funeqi 6557	. . . . . . 7 ⊢ (Fun ^◡(𝐹 ∘ 𝐺) ↔ Fun (^◡𝐺 ∘ ^◡𝐹))
9	6, 8	sylibr 237	. . . . . 6 ⊢ ((Fun ^◡𝐺 ∧ Fun ^◡𝐹) → Fun ^◡(𝐹 ∘ 𝐺))
10	9	ancoms 463	. . . . 5 ⊢ ((Fun ^◡𝐹 ∧ Fun ^◡𝐺) → Fun ^◡(𝐹 ∘ 𝐺))
11	5, 10	anim12i 624	. . . 4 ⊢ (((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) ∧ (Fun ^◡𝐹 ∧ Fun ^◡𝐺)) → ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
12	11	an4s 672	. . 3 ⊢ (((𝐹:𝐶⟶𝐷 ∧ Fun ^◡𝐹) ∧ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺)) → ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
13	1, 2, 12	syl2anb 609	. 2 ⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝐹 ∘ 𝐺):(^◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
14		df-f1 6541	. 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

Syntax hints: → wi 4 ∧ wa 400 ^◡ccnv 5660 “ cima 5664 ∘ ccom 5665 Fun wfun 6530 ⟶wf 6532 –1-1→wf1 6533

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-pr 5404

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-sn 4590 df-pr 4592 df-op 4596 df-br 5110 df-opab 5174 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541

This theorem is referenced by: f1co 6787 f1cof1b 47814

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