Theorem coss1b 38209

Description: Precomposition with a single-valued class (see cocnv2ssid 38207) preserves and reflects the subclass relation on its range. This is a generalization of the fact that surjections are right-cancellable. (Contributed by BJ, 24-Oct-2026.) TODO: prove cocanss2 46177 from it.

Hypothesis

Ref	Expression
coss1b.1	⊢ (𝜑 → (𝐹 ∘ ◡𝐹) ⊆ I )

Assertion

Ref	Expression
coss1b	⊢ (𝜑 → ((𝐻 ∘ 𝐹) ⊆ (𝐾 ∘ 𝐹) ↔ (𝐻 ↾ ran 𝐹) ⊆ (𝐾 ↾ ran 𝐹)))

Proof of Theorem coss1b

Step	Hyp	Ref	Expression
1		coss1 5841	. . 3 ⊢ ((𝐻 ∘ 𝐹) ⊆ (𝐾 ∘ 𝐹) → ((𝐻 ∘ 𝐹) ∘ ◡𝐹) ⊆ ((𝐾 ∘ 𝐹) ∘ ◡𝐹))
2		coass 6288	. . . . 5 ⊢ ((𝐻 ∘ 𝐹) ∘ ◡𝐹) = (𝐻 ∘ (𝐹 ∘ ◡𝐹))
3		coss1b.1	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘ ◡𝐹) ⊆ I )
4		iss 6043	. . . . . . . . 9 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)))
5	3, 4	sylib 221	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)))
6		dmcocnv2 5974	. . . . . . . . 9 ⊢ dom (𝐹 ∘ ◡𝐹) = ran 𝐹
7	6	reseq2i 5981	. . . . . . . 8 ⊢ ( I ↾ dom (𝐹 ∘ ◡𝐹)) = ( I ↾ ran 𝐹)
8	5, 7	eqtrdi 2814	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
9	8	coeq2d 5849	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘ ◡𝐹)) = (𝐻 ∘ ( I ↾ ran 𝐹)))
10		coires1 6286	. . . . . 6 ⊢ (𝐻 ∘ ( I ↾ ran 𝐹)) = (𝐻 ↾ ran 𝐹)
11	9, 10	eqtrdi 2814	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘ ◡𝐹)) = (𝐻 ↾ ran 𝐹))
12	2, 11	eqtrid 2810	. . . 4 ⊢ (𝜑 → ((𝐻 ∘ 𝐹) ∘ ◡𝐹) = (𝐻 ↾ ran 𝐹))
13		coass 6288	. . . . 5 ⊢ ((𝐾 ∘ 𝐹) ∘ ◡𝐹) = (𝐾 ∘ (𝐹 ∘ ◡𝐹))
14	8	coeq2d 5849	. . . . . 6 ⊢ (𝜑 → (𝐾 ∘ (𝐹 ∘ ◡𝐹)) = (𝐾 ∘ ( I ↾ ran 𝐹)))
15		coires1 6286	. . . . . 6 ⊢ (𝐾 ∘ ( I ↾ ran 𝐹)) = (𝐾 ↾ ran 𝐹)
16	14, 15	eqtrdi 2814	. . . . 5 ⊢ (𝜑 → (𝐾 ∘ (𝐹 ∘ ◡𝐹)) = (𝐾 ↾ ran 𝐹))
17	13, 16	eqtrid 2810	. . . 4 ⊢ (𝜑 → ((𝐾 ∘ 𝐹) ∘ ◡𝐹) = (𝐾 ↾ ran 𝐹))
18	12, 17	sseq12d 3946	. . 3 ⊢ (𝜑 → (((𝐻 ∘ 𝐹) ∘ ◡𝐹) ⊆ ((𝐾 ∘ 𝐹) ∘ ◡𝐹) ↔ (𝐻 ↾ ran 𝐹) ⊆ (𝐾 ↾ ran 𝐹)))
19	1, 18	imbitrid 247	. 2 ⊢ (𝜑 → ((𝐻 ∘ 𝐹) ⊆ (𝐾 ∘ 𝐹) → (𝐻 ↾ ran 𝐹) ⊆ (𝐾 ↾ ran 𝐹)))
20		coss1 5841	. . 3 ⊢ ((𝐻 ↾ ran 𝐹) ⊆ (𝐾 ↾ ran 𝐹) → ((𝐻 ↾ ran 𝐹) ∘ 𝐹) ⊆ ((𝐾 ↾ ran 𝐹) ∘ 𝐹))
21		coresrn 38197	. . 3 ⊢ ((𝐻 ↾ ran 𝐹) ∘ 𝐹) = (𝐻 ∘ 𝐹)
22		coresrn 38197	. . 3 ⊢ ((𝐾 ↾ ran 𝐹) ∘ 𝐹) = (𝐾 ∘ 𝐹)
23	20, 21, 22	3sstr3g 3965	. 2 ⊢ ((𝐻 ↾ ran 𝐹) ⊆ (𝐾 ↾ ran 𝐹) → (𝐻 ∘ 𝐹) ⊆ (𝐾 ∘ 𝐹))
24	19, 23	impbid1 228	1 ⊢ (𝜑 → ((𝐻 ∘ 𝐹) ⊆ (𝐾 ∘ 𝐹) ↔ (𝐻 ↾ ran 𝐹) ⊆ (𝐾 ↾ ran 𝐹)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ⊆ wss 3900   I cid 5550  ^◡ccnv 5655  dom cdm 5656  ran crn 5657   ↾ cres 5658   ∘ ccom 5660

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-ext 2735  ax-sep 5253  ax-pr 5396

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  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 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668

This theorem is used by: (None)