Theorem cocnv1ssidim 38205

Description: Consequence of the composition of a converse with a class being included in the identity relation. The instance where the two classes are equal expresses injectivity. (Contributed by BJ, 1-Nov-2026.)

Hypotheses

Ref	Expression
cocnv1ssidim.1	⊢ 𝑋 ∈ V
cocnv1ssidim.2	⊢ 𝑌 ∈ V
cocnv1ssidim.3	⊢ 𝑍 ∈ V

Assertion

Ref	Expression
cocnv1ssidim	⊢ ((◡𝑆 ∘ 𝑅) ⊆ I → ((𝑋𝑅𝑌 ∧ 𝑍𝑆𝑌) → 𝑋 = 𝑍))

Proof of Theorem cocnv1ssidim

Step	Hyp	Ref	Expression
1		cocnv1ssidim.2	. . . 4 ⊢ 𝑌 ∈ V
2		cocnv1ssidim.3	. . . 4 ⊢ 𝑍 ∈ V
3	1, 2	brcnv 5870	. . 3 ⊢ (𝑌◡𝑆𝑍 ↔ 𝑍𝑆𝑌)
4		cocnv1ssidim.1	. . . 4 ⊢ 𝑋 ∈ V
5	4, 1, 2	brcoi 5856	. . 3 ⊢ ((𝑋𝑅𝑌 ∧ 𝑌◡𝑆𝑍) → 𝑋(◡𝑆 ∘ 𝑅)𝑍)
6	3, 5	sylan2br 607	. 2 ⊢ ((𝑋𝑅𝑌 ∧ 𝑍𝑆𝑌) → 𝑋(◡𝑆 ∘ 𝑅)𝑍)
7		ssbr 5153	. . 3 ⊢ ((◡𝑆 ∘ 𝑅) ⊆ I → (𝑋(◡𝑆 ∘ 𝑅)𝑍 → 𝑋 I 𝑍))
8	2	ideq 5838	. . 3 ⊢ (𝑋 I 𝑍 ↔ 𝑋 = 𝑍)
9	7, 8	imbitrdi 254	. 2 ⊢ ((◡𝑆 ∘ 𝑅) ⊆ I → (𝑋(◡𝑆 ∘ 𝑅)𝑍 → 𝑋 = 𝑍))
10	6, 9	syl5 35	1 ⊢ ((◡𝑆 ∘ 𝑅) ⊆ I → ((𝑋𝑅𝑌 ∧ 𝑍𝑆𝑌) → 𝑋 = 𝑍))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   class class class wbr 5106   I cid 5550  ^◡ccnv 5655   ∘ 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

This theorem is used by: (None)