Theorem funeq 5127

Description: Equality theorem for function predicate. (Contributed by set.mm contributors, 16-Aug-1994.)

Assertion

Ref	Expression
funeq	⊢ (A = B → (Fun A ↔ Fun B))

Proof of Theorem funeq

Step	Hyp	Ref	Expression
1		funss 5126	. . . 4 ⊢ (B ⊆ A → (Fun A → Fun B))
2		funss 5126	. . . 4 ⊢ (A ⊆ B → (Fun B → Fun A))
3	1, 2	anim12i 549	. . 3 ⊢ ((B ⊆ A ∧ A ⊆ B) → ((Fun A → Fun B) ∧ (Fun B → Fun A)))
4	3	ancoms 439	. 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → ((Fun A → Fun B) ∧ (Fun B → Fun A)))
5		eqss 3287	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
6		dfbi2 609	. 2 ⊢ ((Fun A ↔ Fun B) ↔ ((Fun A → Fun B) ∧ (Fun B → Fun A)))
7	4, 5, 6	3imtr4i 257	1 ⊢ (A = B → (Fun A ↔ Fun B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ⊆ wss 3257  Fun wfun 4775

This theorem was proved from 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 theorem 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623  df-br 4640  df-co 4726  df-cnv 4785  df-fun 4789

This theorem is referenced by:  funeqi  5128  funeqd  5129  fununi  5160  funcnvuni  5161  cnvresid  5166  fneq1  5173  elfuns  5829  elfunsg  5830  elpmg  6013  fundmeng  6044