Theorem cnvsymOLD 6040

Description: Obsolete proof of cnvsym 6039 as of 29-Dec-2024. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by SN, 23-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem cnvsymOLD

Step	Hyp	Ref	Expression
1		relcnv 6029	. . 3 ⊢ Rel ◡𝑅
2		ssrel3 5715	. . 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		alcom 2155	. 2 ⊢ (∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥))
5		vex 3444	. . . . 5 ⊢ 𝑦 ∈ V
6		vex 3444	. . . . 5 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 5811	. . . 4 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
8	7	imbi1i 349	. . 3 ⊢ ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
9	8	2albii 1821	. 2 ⊢ (∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
10	3, 4, 9	3bitri 296	1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1538   ⊆ wss 3896   class class class wbr 5086  ^◡ccnv 5606  Rel wrel 5612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pr 5366

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3442  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5087  df-opab 5149  df-xp 5613  df-rel 5614  df-cnv 5615

This theorem is referenced by: (None)