Theorem relcnvtrOLD 6259

Description: Obsolete form of relcnvtrg 6257 as of 8-Aug-2026. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Peter Mazsa, 17-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
relcnvtrOLD	⊢ (Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))

Proof of Theorem relcnvtrOLD

Step	Hyp	Ref	Expression
1		3anidm 1121	. 2 ⊢ ((Rel 𝑅 ∧ Rel 𝑅 ∧ Rel 𝑅) ↔ Rel 𝑅)
2		relcnvtrgOLD 6258	. 2 ⊢ ((Rel 𝑅 ∧ Rel 𝑅 ∧ Rel 𝑅) → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))
3	1, 2	sylbir 238	1 ⊢ (Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   ⊆ wss 3898  ^◡ccnv 5646   ∘ ccom 5651  Rel wrel 5652

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-ext 2732  ax-sep 5248  ax-pr 5390

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 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656

This theorem is used by: (None)