Theorem foeq1 3663

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 3578	. . 3 |- (F = G -> (F Fn A <-> G Fn A))
2		rneq 3335	. . . 4 |- (F = G -> ran F = ran G)
3	2	eqeq1d 1481	. . 3 |- (F = G -> (ran F = B <-> ran G = B))
4	1, 3	anbi12d 627	. 2 |- (F = G -> ((F Fn A /\ ran F = B) <-> (G Fn A /\ ran G = B)))
5		df-fo 3192	. 2 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
6		df-fo 3192	. 2 |- (G:A-onto->B <-> (G Fn A /\ ran G = B))
7	4, 5, 6	3bitr4g 554	1 |- (F = G -> (F:A-onto->B <-> G:A-onto->B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955  ran crn 3167   Fn wfn 3173  -onto->wfo 3176

This theorem is referenced by:  f1oeq1 3679  exfo 3817  fo1st 4084  fo2nd 4085  fodomr 4472  fodomfi 4549  ruclem39 7508  infmap2lem1 7539  pjfot 9608  elunopt 9756  elunop2t 9894

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-id 2831  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-fun 3188  df-fn 3189  df-fo 3192

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