Theorem bnj137 12469

Description: First-order logic and set theory.

Assertion

Ref	Expression
bnj137	|- (F Fn A -> (A = (/) <-> F = (/)))

Proof of Theorem bnj137

Step	Hyp	Ref	Expression
1		fnresdisj 4523	. . 3 |- (F Fn A -> ((A i^i A) = (/) <-> (F |` A) = (/)))
2		inidm 2803	. . . 4 |- (A i^i A) = A
3	2	eqeq1i 1891	. . 3 |- ((A i^i A) = (/) <-> A = (/))
4	1, 3	syl5bbr 593	. 2 |- (F Fn A -> (A = (/) <-> (F |` A) = (/)))
5		fnresdm 4522	. . 3 |- (F Fn A -> (F |` A) = F)
6	5	eqeq1d 1892	. 2 |- (F Fn A -> ((F |` A) = (/) <-> F = (/)))
7	4, 6	bitrd 587	1 |- (F Fn A -> (A = (/) <-> F = (/)))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   i^i cin 2592  (/)c0 2875   |` cres 3988   Fn wfn 3993

This theorem is referenced by:  bnj138 12470

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-xp 4000  df-rel 4001  df-dm 4004  df-res 4006  df-fun 4008  df-fn 4009

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