Theorem symreleq 38062

Description: Equality theorem for symmetric relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.)

Assertion

Ref	Expression
symreleq	⊢ (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))

Proof of Theorem symreleq

Step	Hyp	Ref	Expression
1		cnveq 5880	. . . 4 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆)
2		id 22	. . . 4 ⊢ (𝑅 = 𝑆 → 𝑅 = 𝑆)
3	1, 2	sseq12d 4015	. . 3 ⊢ (𝑅 = 𝑆 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑆 ⊆ 𝑆))
4		releq 5782	. . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
5	3, 4	anbi12d 630	. 2 ⊢ (𝑅 = 𝑆 → ((◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅) ↔ (◡𝑆 ⊆ 𝑆 ∧ Rel 𝑆)))
6		dfsymrel2 38053	. 2 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))
7		dfsymrel2 38053	. 2 ⊢ ( SymRel 𝑆 ↔ (◡𝑆 ⊆ 𝑆 ∧ Rel 𝑆))
8	5, 6, 7	3bitr4g 313	1 ⊢ (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ⊆ wss 3949  ^◡ccnv 5681  Rel wrel 5687   SymRel wsymrel 37693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-symrel 38048

This theorem is referenced by:  eqvreleq  38106