Theorem symreleq 38549

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 5837	. . . 4 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆)
2		id 22	. . . 4 ⊢ (𝑅 = 𝑆 → 𝑅 = 𝑆)
3	1, 2	sseq12d 3980	. . 3 ⊢ (𝑅 = 𝑆 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑆 ⊆ 𝑆))
4		releq 5739	. . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
5	3, 4	anbi12d 632	. 2 ⊢ (𝑅 = 𝑆 → ((◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅) ↔ (◡𝑆 ⊆ 𝑆 ∧ Rel 𝑆)))
6		dfsymrel2 38540	. 2 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))
7		dfsymrel2 38540	. 2 ⊢ ( SymRel 𝑆 ↔ (◡𝑆 ⊆ 𝑆 ∧ Rel 𝑆))
8	5, 6, 7	3bitr4g 314	1 ⊢ (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ⊆ wss 3914  ^◡ccnv 5637  Rel wrel 5643   SymRel wsymrel 38181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-symrel 38535

This theorem is referenced by:  eqvreleq  38593