f1coOLD - Metamath Proof Explorer

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Theorem f1coOLD 6811

Description: Obsolete version of f1co 6810 as of 20-Sep-2024. (Contributed by NM, 28-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)

**Assertion**
Ref	Expression
f1coOLD	⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶)

**Proof of Theorem f1coOLD**
Step	Hyp	Ref	Expression
1		df-f1 6558	. . 3 ⊢ (𝐹:𝐵–1-1→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ Fun ^◡𝐹))
2		df-f1 6558	. . 3 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺))
3		fco 6752	. . . . 5 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
4		funco 6598	. . . . . . 7 ⊢ ((Fun ^◡𝐺 ∧ Fun ^◡𝐹) → Fun (^◡𝐺 ∘ ^◡𝐹))
5		cnvco 5892	. . . . . . . 8 ⊢ ^◡(𝐹 ∘ 𝐺) = (^◡𝐺 ∘ ^◡𝐹)
6	5	funeqi 6579	. . . . . . 7 ⊢ (Fun ^◡(𝐹 ∘ 𝐺) ↔ Fun (^◡𝐺 ∘ ^◡𝐹))
7	4, 6	sylibr 233	. . . . . 6 ⊢ ((Fun ^◡𝐺 ∧ Fun ^◡𝐹) → Fun ^◡(𝐹 ∘ 𝐺))
8	7	ancoms 457	. . . . 5 ⊢ ((Fun ^◡𝐹 ∧ Fun ^◡𝐺) → Fun ^◡(𝐹 ∘ 𝐺))
9	3, 8	anim12i 611	. . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) ∧ (Fun ^◡𝐹 ∧ Fun ^◡𝐺)) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
10	9	an4s 658	. . 3 ⊢ (((𝐹:𝐵⟶𝐶 ∧ Fun ^◡𝐹) ∧ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺)) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
11	1, 2, 10	syl2anb 596	. 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
12		df-f1 6558	. 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ^◡(𝐹 ∘ 𝐺)))
13	11, 12	sylibr 233	1 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ^◡ccnv 5681 ∘ ccom 5686 Fun wfun 6547 ⟶wf 6549 –1-1→wf1 6550

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558

This theorem is referenced by: (None)

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