cocanss1 - Mathbox for Eric Schmidt

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Theorem cocanss1 45923

Description: Cancellation law for composition. Suggested by BJ. (Contributed by Eric Schmidt, 30-Sep-2026.)

**Hypotheses**
Ref	Expression
cocan1g.1	⊢ (𝜑 → Fun ^◡𝐹)
cocan1g.2	⊢ (𝜑 → Rel 𝐻)
cocan1g.3	⊢ (𝜑 → ran 𝐻 ⊆ dom 𝐹)

**Assertion**
Ref	Expression
cocanss1	⊢ (𝜑 → ((𝐹 ∘ 𝐻) ⊆ (𝐹 ∘ 𝐾) ↔ 𝐻 ⊆ 𝐾))

**Proof of Theorem cocanss1**
Step	Hyp	Ref	Expression
1		cocan1g.1	. . 3 ⊢ (𝜑 → Fun ^◡𝐹)
2		relcnv 6100	. . . 4 ⊢ Rel ^◡𝐻
3	2	a1i 11	. . 3 ⊢ (𝜑 → Rel ^◡𝐻)
4		cocan1g.3	. . . 4 ⊢ (𝜑 → ran 𝐻 ⊆ dom 𝐹)
5		df-rn 5662	. . . 4 ⊢ ran 𝐻 = dom ^◡𝐻
6		dfdm4 5877	. . . 4 ⊢ dom 𝐹 = ran ^◡𝐹
7	4, 5, 6	3sstr3g 3983	. . 3 ⊢ (𝜑 → dom ^◡𝐻 ⊆ ran ^◡𝐹)
8	1, 3, 7	cocanss2 45921	. 2 ⊢ (𝜑 → ((^◡𝐻 ∘ ^◡𝐹) ⊆ (^◡𝐾 ∘ ^◡𝐹) ↔ ^◡𝐻 ⊆ ^◡𝐾))
9		relco 6104	. . . . 5 ⊢ Rel (𝐹 ∘ 𝐻)
10		cnvssb 6189	. . . . 5 ⊢ (Rel (𝐹 ∘ 𝐻) → ((𝐹 ∘ 𝐻) ⊆ (𝐹 ∘ 𝐾) ↔ ^◡(𝐹 ∘ 𝐻) ⊆ ^◡(𝐹 ∘ 𝐾)))
11	9, 10	ax-mp 5	. . . 4 ⊢ ((𝐹 ∘ 𝐻) ⊆ (𝐹 ∘ 𝐾) ↔ ^◡(𝐹 ∘ 𝐻) ⊆ ^◡(𝐹 ∘ 𝐾))
12		cnvco 5867	. . . . 5 ⊢ ^◡(𝐹 ∘ 𝐻) = (^◡𝐻 ∘ ^◡𝐹)
13		cnvco 5867	. . . . 5 ⊢ ^◡(𝐹 ∘ 𝐾) = (^◡𝐾 ∘ ^◡𝐹)
14	12, 13	sseq12i 3961	. . . 4 ⊢ (^◡(𝐹 ∘ 𝐻) ⊆ ^◡(𝐹 ∘ 𝐾) ↔ (^◡𝐻 ∘ ^◡𝐹) ⊆ (^◡𝐾 ∘ ^◡𝐹))
15	11, 14	bitri 278	. . 3 ⊢ ((𝐹 ∘ 𝐻) ⊆ (𝐹 ∘ 𝐾) ↔ (^◡𝐻 ∘ ^◡𝐹) ⊆ (^◡𝐾 ∘ ^◡𝐹))
16	15	a1i 11	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ⊆ (𝐹 ∘ 𝐾) ↔ (^◡𝐻 ∘ ^◡𝐹) ⊆ (^◡𝐾 ∘ ^◡𝐹)))
17		cocan1g.2	. . 3 ⊢ (𝜑 → Rel 𝐻)
18		cnvssb 6189	. . 3 ⊢ (Rel 𝐻 → (𝐻 ⊆ 𝐾 ↔ ^◡𝐻 ⊆ ^◡𝐾))
19	17, 18	syl 18	. 2 ⊢ (𝜑 → (𝐻 ⊆ 𝐾 ↔ ^◡𝐻 ⊆ ^◡𝐾))
20	8, 16, 19	3bitr4d 314	1 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ⊆ (𝐹 ∘ 𝐾) ↔ 𝐻 ⊆ 𝐾))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ⊆ wss 3899 ^◡ccnv 5650 dom cdm 5651 ran crn 5652 ∘ ccom 5655 Rel wrel 5656 Fun wfun 6532

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-12 2213 ax-ext 2733 ax-sep 5249 ax-pr 5391

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-sb 2100 df-clab 2740 df-cleq 2753 df-clel 2836 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 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 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-fun 6540

This theorem is used by: cocan1g 45924

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