Theorem cnvsym 6102

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 2194. (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 6094	. . 3 ⊢ Rel ◡𝑅
2		ssrel3 5758	. . 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 347	. . 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 347	. . 3 ⊢ (𝑥 = 𝑧 → ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑦◡𝑅𝑧 → 𝑦𝑅𝑧)))
10	6, 9	alcomw 2078	. 2 ⊢ (∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥))
11		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
12		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
13	11, 12	brcnv 5856	. . . 4 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
14	13	imbi1i 352	. . 3 ⊢ ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
15	14	2albii 1853	. 2 ⊢ (∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
16	3, 10, 15	3bitri 300	1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   ⊆ wss 3898   class class class wbr 5102  ^◡ccnv 5646  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

This theorem is used by:  dfer2  8696  relcnveq3  39179  relcnveq  39180  relcnveq2  39181  cnvcosseq  39379  symrelcoss2  39408  dfsymrels3  39478  elrelscnveq3  39479  elrelscnveq  39480  elrelscnveq2  39481  dfsymrel3  39486  symrefref3  39500  refsymrels3  39502  elrefsymrels3  39506  dfeqvrels3  39525