Theorem feq1 4973

Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
feq1	|- ( F = G -> ( F :A -->B <-> G :A -->B))

Proof of Theorem feq1

Step	Hyp	Ref	Expression
1		fneq1 4930	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A))
2		rneq 4504	. . . 4 |- ( F = G -> ran F = ran G)
3	2	sseq1d 2966	. . 3 |- ( F = G -> ( ran F C_ B <-> ran G C_ B))
4	1, 3	anbi12d 442	. 2 |- ( F = G -> (( F Fn A /\ ran F C_ B) <-> ( G Fn A /\ ran G C_ B)))
5		df-f 4849	. 2 |- ( F :A -->B <-> ( F Fn A /\ ran F C_ B))
6		df-f 4849	. 2 |- ( G :A -->B <-> ( G Fn A /\ ran G C_ B))
7	4, 5, 6	3bitr4g 212	1 |- ( F = G -> ( F :A -->B <-> G :A -->B))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242   C_ wss 2911   ran crn 4289   Fn wfn 4840   -->wf 4841

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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849

This theorem is referenced by:  feq1d  4977  feq1i  4982  f00  5024  fconstg  5026  f1eq1  5030  fconst2g  5319  1fv  8766