Theorem coss2b 38211

Description: Postcomposition with an injective class (see cocnv1ssid 38204) preserves and reflects the subclass relation up to corestriction to its domain. This is a generalization of the fact that injections are left-cancellable. Compare with coss1b 38209. There is no special syntax for corestriction as explained in the comment of df-res 5668. (Contributed by BJ, 24-Oct-2026.) TODO: prove cocanss1 46179 from it.

Hypothesis

Ref	Expression
coss2b.1	⊢ (𝜑 → (◡𝐹 ∘ 𝐹) ⊆ I )

Assertion

Ref	Expression
coss2b	⊢ (𝜑 → ((𝐹 ∘ 𝐻) ⊆ (𝐹 ∘ 𝐾) ↔ (𝐻 ∩ (V × dom 𝐹)) ⊆ (𝐾 ∩ (V × dom 𝐹))))

Proof of Theorem coss2b

Step	Hyp	Ref	Expression
1		coss2 5842	. . 3 ⊢ ((𝐹 ∘ 𝐻) ⊆ (𝐹 ∘ 𝐾) → (◡𝐹 ∘ (𝐹 ∘ 𝐻)) ⊆ (◡𝐹 ∘ (𝐹 ∘ 𝐾)))
2		coass 6288	. . . . 5 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐻) = (◡𝐹 ∘ (𝐹 ∘ 𝐻))
3		coss2b.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		dmcocnv1 5973	. . . . . . . . 9 ⊢ dom (◡𝐹 ∘ 𝐹) = dom 𝐹
7	6	reseq2i 5981	. . . . . . . 8 ⊢ ( I ↾ dom (◡𝐹 ∘ 𝐹)) = ( I ↾ dom 𝐹)
8	5, 7	eqtrdi 2814	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 ∘ 𝐹) = ( I ↾ dom 𝐹))
9	8	coeq1d 5848	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐻) = (( I ↾ dom 𝐹) ∘ 𝐻))
10		coires2 6287	. . . . . 6 ⊢ (( I ↾ dom 𝐹) ∘ 𝐻) = (𝐻 ∩ (V × dom 𝐹))
11	9, 10	eqtrdi 2814	. . . . 5 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐻) = (𝐻 ∩ (V × dom 𝐹)))
12	2, 11	eqtr3id 2812	. . . 4 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐻)) = (𝐻 ∩ (V × dom 𝐹)))
13		coass 6288	. . . . 5 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐾) = (◡𝐹 ∘ (𝐹 ∘ 𝐾))
14	8	coeq1d 5848	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐾) = (( I ↾ dom 𝐹) ∘ 𝐾))
15		coires2 6287	. . . . . 6 ⊢ (( I ↾ dom 𝐹) ∘ 𝐾) = (𝐾 ∩ (V × dom 𝐹))
16	14, 15	eqtrdi 2814	. . . . 5 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐾) = (𝐾 ∩ (V × dom 𝐹)))
17	13, 16	eqtr3id 2812	. . . 4 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐾)) = (𝐾 ∩ (V × dom 𝐹)))
18	12, 17	sseq12d 3946	. . 3 ⊢ (𝜑 → ((◡𝐹 ∘ (𝐹 ∘ 𝐻)) ⊆ (◡𝐹 ∘ (𝐹 ∘ 𝐾)) ↔ (𝐻 ∩ (V × dom 𝐹)) ⊆ (𝐾 ∩ (V × dom 𝐹))))
19	1, 18	imbitrid 247	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ⊆ (𝐹 ∘ 𝐾) → (𝐻 ∩ (V × dom 𝐹)) ⊆ (𝐾 ∩ (V × dom 𝐹))))
20		coss2 5842	. . 3 ⊢ ((𝐻 ∩ (V × dom 𝐹)) ⊆ (𝐾 ∩ (V × dom 𝐹)) → (𝐹 ∘ (𝐻 ∩ (V × dom 𝐹))) ⊆ (𝐹 ∘ (𝐾 ∩ (V × dom 𝐹))))
21		cnvrescnv 6204	. . . . 5 ⊢ ◡(◡𝐻 ↾ dom 𝐹) = (𝐻 ∩ (V × dom 𝐹))
22	21	coeq2i 5847	. . . 4 ⊢ (𝐹 ∘ ◡(◡𝐻 ↾ dom 𝐹)) = (𝐹 ∘ (𝐻 ∩ (V × dom 𝐹)))
23		ssid 3935	. . . . 5 ⊢ dom 𝐹 ⊆ dom 𝐹
24		cores2 6278	. . . . 5 ⊢ (dom 𝐹 ⊆ dom 𝐹 → (𝐹 ∘ ◡(◡𝐻 ↾ dom 𝐹)) = (𝐹 ∘ 𝐻))
25	23, 24	ax-mp 5	. . . 4 ⊢ (𝐹 ∘ ◡(◡𝐻 ↾ dom 𝐹)) = (𝐹 ∘ 𝐻)
26	22, 25	eqtr3i 2788	. . 3 ⊢ (𝐹 ∘ (𝐻 ∩ (V × dom 𝐹))) = (𝐹 ∘ 𝐻)
27		cnvrescnv 6204	. . . . 5 ⊢ ◡(◡𝐾 ↾ dom 𝐹) = (𝐾 ∩ (V × dom 𝐹))
28	27	coeq2i 5847	. . . 4 ⊢ (𝐹 ∘ ◡(◡𝐾 ↾ dom 𝐹)) = (𝐹 ∘ (𝐾 ∩ (V × dom 𝐹)))
29		cores2 6278	. . . . 5 ⊢ (dom 𝐹 ⊆ dom 𝐹 → (𝐹 ∘ ◡(◡𝐾 ↾ dom 𝐹)) = (𝐹 ∘ 𝐾))
30	23, 29	ax-mp 5	. . . 4 ⊢ (𝐹 ∘ ◡(◡𝐾 ↾ dom 𝐹)) = (𝐹 ∘ 𝐾)
31	28, 30	eqtr3i 2788	. . 3 ⊢ (𝐹 ∘ (𝐾 ∩ (V × dom 𝐹))) = (𝐹 ∘ 𝐾)
32	20, 26, 31	3sstr3g 3965	. 2 ⊢ ((𝐻 ∩ (V × dom 𝐹)) ⊆ (𝐾 ∩ (V × dom 𝐹)) → (𝐹 ∘ 𝐻) ⊆ (𝐹 ∘ 𝐾))
33	19, 32	impbid1 228	1 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ⊆ (𝐹 ∘ 𝐾) ↔ (𝐻 ∩ (V × dom 𝐹)) ⊆ (𝐾 ∩ (V × dom 𝐹))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   I cid 5550   × cxp 5654  ^◡ccnv 5655  dom cdm 5656   ↾ 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)