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Theorem cnvsym 6075

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 2163. (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 6067	. . 3 ⊢ Rel ^◡𝑅
2		ssrel3 5739	. . 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 5089	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦^◡𝑅𝑥 ↔ 𝑧^◡𝑅𝑥))
5		breq1 5089	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
6	4, 5	imbi12d 344	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑧^◡𝑅𝑥 → 𝑧𝑅𝑥)))
7		breq2 5090	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦^◡𝑅𝑥 ↔ 𝑦^◡𝑅𝑧))
8		breq2 5090	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
9	7, 8	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑧 → ((𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑦^◡𝑅𝑧 → 𝑦𝑅𝑧)))
10	6, 9	alcomw 2047	. 2 ⊢ (∀𝑦∀𝑥(𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥))
11		vex 3434	. . . . 5 ⊢ 𝑦 ∈ V
12		vex 3434	. . . . 5 ⊢ 𝑥 ∈ V
13	11, 12	brcnv 5835	. . . 4 ⊢ (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
14	13	imbi1i 349	. . 3 ⊢ ((𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
15	14	2albii 1822	. 2 ⊢ (∀𝑥∀𝑦(𝑦^◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
16	3, 10, 15	3bitri 297	1 ⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∀wal 1540 ⊆ wss 3890 class class class wbr 5086 ^◡ccnv 5627 Rel wrel 5633

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-ext 2709 ax-sep 5232 ax-pr 5374

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-sb 2069 df-clab 2716 df-cleq 2729 df-clel 2812 df-rab 3391 df-v 3432 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-br 5087 df-opab 5149 df-xp 5634 df-rel 5635 df-cnv 5636

This theorem is referenced by: dfer2 8641 relcnveq3 38670 relcnveq 38671 relcnveq2 38672 cnvcosseq 38870 symrelcoss2 38899 dfsymrels3 38969 elrelscnveq3 38970 elrelscnveq 38971 elrelscnveq2 38972 dfsymrel3 38977 symrefref3 38991 refsymrels3 38993 elrefsymrels3 38997 dfeqvrels3 39016

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