Theorem cnvsym 6103

Description: Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by SN, 23-Dec-2024.) Avoid ax-11 2193. (Revised by BTernaryTau, 29-Dec-2024.)

Assertion

Ref	Expression
cnvsym	⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem cnvsym

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 6095	. . 3 ⊢ Rel ◡𝑅
2		ssrel3 5760	. . 3 ⊢ (Rel ◡𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥)))
3	1, 2	ax-mp 5	. 2 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥))
4		breq1 5105	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦◡𝑅𝑥 ↔ 𝑧◡𝑅𝑥))
5		breq1 5105	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
6	4, 5	imbi12d 346	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑧◡𝑅𝑥 → 𝑧𝑅𝑥)))
7		breq2 5106	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦◡𝑅𝑥 ↔ 𝑦◡𝑅𝑧))
8		breq2 5106	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
9	7, 8	imbi12d 346	. . 3 ⊢ (𝑥 = 𝑧 → ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑦◡𝑅𝑧 → 𝑦𝑅𝑧)))
10	6, 9	alcomw 2067	. 2 ⊢ (∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥))
11		vex 3460	. . . . 5 ⊢ 𝑦 ∈ V
12		vex 3460	. . . . 5 ⊢ 𝑥 ∈ V
13	11, 12	brcnv 5856	. . . 4 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
14	13	imbi1i 351	. . 3 ⊢ ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
15	14	2albii 1842	. 2 ⊢ (∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
16	3, 10, 15	3bitri 299	1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1560   ⊆ wss 3906   class class class wbr 5102  ^◡ccnv 5648  Rel wrel 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657

This theorem is referenced by:  dfer2  8681  relcnveq3  38831  relcnveq  38832  relcnveq2  38833  cnvcosseq  39031  symrelcoss2  39060  dfsymrels3  39130  elrelscnveq3  39131  elrelscnveq  39132  elrelscnveq2  39133  dfsymrel3  39138  symrefref3  39152  refsymrels3  39154  elrefsymrels3  39158  dfeqvrels3  39177