Theorem cnveq 4887

Description: Equality theorem for converse. (Contributed by set.mm contributors, 13-Aug-1995.)

Assertion

Ref	Expression
cnveq	⊢ (A = B → ◡A = ◡B)

Proof of Theorem cnveq

Step	Hyp	Ref	Expression
1		cnvss 4886	. . 3 ⊢ (A ⊆ B → ◡A ⊆ ◡B)
2		cnvss 4886	. . 3 ⊢ (B ⊆ A → ◡B ⊆ ◡A)
3	1, 2	anim12i 549	. 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → (◡A ⊆ ◡B ∧ ◡B ⊆ ◡A))
4		eqss 3288	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
5		eqss 3288	. 2 ⊢ (◡A = ◡B ↔ (◡A ⊆ ◡B ∧ ◡B ⊆ ◡A))
6	3, 4, 5	3imtr4i 257	1 ⊢ (A = B → ◡A = ◡B)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   = wceq 1642   ⊆ wss 3258  ^◡ccnv 4772

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-br 4641  df-cnv 4786

This theorem is used by:  cnveqi  4888  cnveqd  4889  cnveqb  5064  funcnvuni  5162  f1eq1  5254  f1o00  5318  enprmaplem3  6079  enprmaplem5  6081  enprmaplem6  6082