Theorem bnj134 12478

Description: First-order logic and set theory.

Hypotheses

Ref	Expression
bnj134.1	|- A e. _V
bnj134.2	|- B e. _V

Assertion

Ref	Expression
bnj134	|- ((F Fn {A} /\ (F` A) = B) <-> F = {<.A, B>.})

Proof of Theorem bnj134

Step	Hyp	Ref	Expression
1		ancom 482	. 2 |- ((F Fn {A} /\ (F` A) = B) <-> ((F` A) = B /\ F Fn {A}))
2		elsni 3066	. . . 4 |- ((F` A) e. {B} -> (F` A) = B)
3		bnj134.1	. . . . . 6 |- A e. _V
4		fvex 4689	. . . . . 6 |- (F` A) e. _V
5	3, 4	fsn 4807	. . . . 5 |- (F:{A}-->{(F` A)} <-> F = {<.A, (F` A)>.})
6		ffn 4562	. . . . 5 |- (F:{A}-->{(F` A)} -> F Fn {A})
7	5, 6	sylbir 218	. . . 4 |- (F = {<.A, (F` A)>.} -> F Fn {A})
8	2, 7	anim12i 360	. . 3 |- (((F` A) e. {B} /\ F = {<.A, (F` A)>.}) -> ((F` A) = B /\ F Fn {A}))
9		bnj134.2	. . . . . 6 |- B e. _V
10	9	elsnc2 3071	. . . . 5 |- ((F` A) e. {B} <-> (F` A) = B)
11	10	biimpri 169	. . . 4 |- ((F` A) = B -> (F` A) e. {B})
12	3, 4	bnj145 12477	. . . 4 |- (F Fn {A} -> F = {<.A, (F` A)>.})
13	11, 12	anim12i 360	. . 3 |- (((F` A) = B /\ F Fn {A}) -> ((F` A) e. {B} /\ F = {<.A, (F` A)>.}))
14	8, 13	impbii 174	. 2 |- (((F` A) e. {B} /\ F = {<.A, (F` A)>.}) <-> ((F` A) = B /\ F Fn {A}))
15	3	fsn2 4809	. . 3 |- (F:{A}-->{B} <-> ((F` A) e. {B} /\ F = {<.A, (F` A)>.}))
16	3, 9	fsn 4807	. . 3 |- (F:{A}-->{B} <-> F = {<.A, B>.})
17	15, 16	bitr3i 192	. 2 |- (((F` A) e. {B} /\ F = {<.A, (F` A)>.}) <-> F = {<.A, B>.})
18	1, 14, 17	3bitr2i 196	1 |- ((F Fn {A} /\ (F` A) = B) <-> F = {<.A, B>.})

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  {csn 3044  <.cop 3046   Fn wfn 3993  -->wf 3994  ` cfv 3998

This theorem is referenced by:  bnj146 12479  bnj147 12480

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-3an 860  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-reu 2111  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014

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