Theorem bnj141 12473

Description: First-order logic and set theory.

Assertion

Ref	Expression
bnj141	|- (F Fn {A} <-> F = {<.x, y>. | (x = A /\ y = (F` A))})

Distinct variable groups:   x,A,y   x,F,y

Proof of Theorem bnj141

Step	Hyp	Ref	Expression
1		dffn5 4717	. 2 |- (F Fn {A} <-> F = {<.x, y>. | (x e. {A} /\ y = (F` x))})
2		elsn 3058	. . . . . 6 |- (x e. {A} <-> x = A)
3	2	anbi1i 539	. . . . 5 |- ((x e. {A} /\ y = (F` x)) <-> (x = A /\ y = (F` x)))
4		fveq2 4681	. . . . . . 7 |- (x = A -> (F` x) = (F` A))
5	4	eqeq2d 1895	. . . . . 6 |- (x = A -> (y = (F` x) <-> y = (F` A)))
6	5	pm5.32i 707	. . . . 5 |- ((x = A /\ y = (F` x)) <-> (x = A /\ y = (F` A)))
7	3, 6	bitri 190	. . . 4 |- ((x e. {A} /\ y = (F` x)) <-> (x = A /\ y = (F` A)))
8	7	opabbii 3402	. . 3 |- {<.x, y>. | (x e. {A} /\ y = (F` x))} = {<.x, y>. | (x = A /\ y = (F` A))}
9	8	eqeq2i 1894	. 2 |- (F = {<.x, y>. | (x e. {A} /\ y = (F` x))} <-> F = {<.x, y>. | (x = A /\ y = (F` A))})
10	1, 9	bitri 190	1 |- (F Fn {A} <-> F = {<.x, y>. | (x = A /\ y = (F` A))})

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {csn 3044  {copab 3395   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  bnj142 12474

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-13 1311  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  ax-un 3790

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-ral 2109  df-rex 2110  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-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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