Theorem funeq 5128

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 5127	. . . 4 ⊢ (B ⊆ A → (Fun A → Fun B))
2		funss 5127	. . . 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 3288	. 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 3258  Fun wfun 4776

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-br 4641  df-co 4727  df-cnv 4786  df-fun 4790

This theorem is referenced by:  funeqi  5129  funeqd  5130  fununi  5161  funcnvuni  5162  cnvresid  5167  fneq1  5174  elfuns  5830  elfunsg  5831  elpmg  6014  fundmeng  6045