Theorem funeq 4864

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

Assertion

Ref	Expression
funeq	|- (A = B -> ( Fun A <-> Fun B))

Proof of Theorem funeq

Step	Hyp	Ref	Expression
1		eqimss2 2992	. . 3 |- (A = B -> B C_ A)
2		funss 4863	. . 3 |- (B C_ A -> ( Fun A -> Fun B))
3	1, 2	syl 14	. 2 |- (A = B -> ( Fun A -> Fun B))
4		eqimss 2991	. . 3 |- (A = B -> A C_ B)
5		funss 4863	. . 3 |- (A C_ B -> ( Fun B -> Fun A))
6	4, 5	syl 14	. 2 |- (A = B -> ( Fun B -> Fun A))
7	3, 6	impbid 120	1 |- (A = B -> ( Fun A <-> Fun B))

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1242   C_ wss 2911   Fun wfun 4839

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-fun 4847

This theorem is referenced by:  funeqi  4865  funeqd  4866  fununi  4910  funcnvuni  4911  cnvresid  4916  fneq1  4930  fundmeng  6223