Theorem foeq1 4613

Description: Equality theorem for onto functions.

Assertion

Ref	Expression
foeq1	|- (F = G -> (F:A-onto->B <-> G:A-onto->B))

Proof of Theorem foeq1

Step	Hyp	Ref	Expression
1		fneq1 4503	. . 3 |- (F = G -> (F Fn A <-> G Fn A))
2		rneq 4186	. . . 4 |- (F = G -> ran F = ran G)
3	2	eqeq1d 1892	. . 3 |- (F = G -> (ran F = B <-> ran G = B))
4	1, 3	anbi12d 690	. 2 |- (F = G -> ((F Fn A /\ ran F = B) <-> (G Fn A /\ ran G = B)))
5		df-fo 4012	. 2 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
6		df-fo 4012	. 2 |- (G:A-onto->B <-> (G Fn A /\ ran G = B))
7	4, 5, 6	3bitr4g 614	1 |- (F = G -> (F:A-onto->B <-> G:A-onto->B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298  ran crn 3987   Fn wfn 3993  -onto->wfo 3996

This theorem is referenced by:  f1oeq1 4630  resdif 4659  exfo 4795  fo1st 5032  fo2nd 5033  fodomr 5547  fodomfi 5656  ruclem39 8817  infmap2lem1 8848  pjfo 11286  elunop 11436  elunop2 11575

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009  df-fo 4012

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