Theorem foeq1 3653

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 3568	. . 3 |- (F = G -> (F Fn A <-> G Fn A))
2		rneq 3328	. . . 4 |- (F = G -> ran F = ran G)
3	2	eqeq1d 1475	. . 3 |- (F = G -> (ran F = B <-> ran G = B))
4	1, 3	anbi12d 626	. 2 |- (F = G -> ((F Fn A /\ ran F = B) <-> (G Fn A /\ ran G = B)))
5		df-fo 3186	. 2 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
6		df-fo 3186	. 2 |- (G:A-onto->B <-> (G Fn A /\ ran G = B))
7	4, 5, 6	3bitr4g 553	1 |- (F = G -> (F:A-onto->B <-> G:A-onto->B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953  ran crn 3161   Fn wfn 3167  -onto->wfo 3170

This theorem is referenced by:  f1oeq1 3669  exfo 3807  fo1st 4075  fo2nd 4076  fodomr 4463  fodomfi 4540  ruclem39 7491  infmap2lem1 7521  pjfot 9568  elunopt 9716  elunop2t 9853

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-fun 3182  df-fn 3183  df-fo 3186

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