Theorem cnvsym 6118

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 2199. (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 6110	. . 3 ⊢ Rel ◡𝑅
2		ssrel3 5776	. . 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 5117	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦◡𝑅𝑥 ↔ 𝑧◡𝑅𝑥))
5		breq1 5117	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
6	4, 5	imbi12d 347	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑧◡𝑅𝑥 → 𝑧𝑅𝑥)))
7		breq2 5118	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦◡𝑅𝑥 ↔ 𝑦◡𝑅𝑧))
8		breq2 5118	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
9	7, 8	imbi12d 347	. . 3 ⊢ (𝑥 = 𝑧 → ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑦◡𝑅𝑧 → 𝑦𝑅𝑧)))
10	6, 9	alcomw 2073	. 2 ⊢ (∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥))
11		vex 3466	. . . . 5 ⊢ 𝑦 ∈ V
12		vex 3466	. . . . 5 ⊢ 𝑥 ∈ V
13	11, 12	brcnv 5872	. . . 4 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
14	13	imbi1i 352	. . 3 ⊢ ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
15	14	2albii 1848	. 2 ⊢ (∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
16	3, 10, 15	3bitri 300	1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1566   ⊆ wss 3913   class class class wbr 5114  ^◡ccnv 5664  Rel wrel 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-rel 5672  df-cnv 5673

This theorem is referenced by:  dfer2  8698  relcnveq3  38926  relcnveq  38927  relcnveq2  38928  cnvcosseq  39126  symrelcoss2  39155  dfsymrels3  39225  elrelscnveq3  39226  elrelscnveq  39227  elrelscnveq2  39228  dfsymrel3  39233  symrefref3  39247  refsymrels3  39249  elrefsymrels3  39253  dfeqvrels3  39272