Theorem bnj148 12481

Description: First-order logic and set theory.

Assertion

Ref	Expression
bnj148	|- ((A e. _V /\ B e. _V) -> ((F Fn {A} /\ (F` A) = B) <-> F = {<.A, B>.}))

Proof of Theorem bnj148

Step	Hyp	Ref	Expression
1		sneq 3054	. . . . . . 7 |- (A = if(A e. _V, A, (/)) -> {A} = {if(A e. _V, A, (/))})
2	1	fneq2d 4506	. . . . . 6 |- (A = if(A e. _V, A, (/)) -> (F Fn {A} <-> F Fn {if(A e. _V, A, (/))}))
3		fveq2 4681	. . . . . . 7 |- (A = if(A e. _V, A, (/)) -> (F` A) = (F` if(A e. _V, A, (/))))
4	3	eqeq1d 1892	. . . . . 6 |- (A = if(A e. _V, A, (/)) -> ((F` A) = B <-> (F` if(A e. _V, A, (/))) = B))
5	2, 4	anbi12d 690	. . . . 5 |- (A = if(A e. _V, A, (/)) -> ((F Fn {A} /\ (F` A) = B) <-> (F Fn {if(A e. _V, A, (/))} /\ (F` if(A e. _V, A, (/))) = B)))
6		opeq1 3158	. . . . . . 7 |- (A = if(A e. _V, A, (/)) -> <.A, B>. = <.if(A e. _V, A, (/)), B>.)
7	6	sneqd 3056	. . . . . 6 |- (A = if(A e. _V, A, (/)) -> {<.A, B>.} = {<.if(A e. _V, A, (/)), B>.})
8	7	eqeq2d 1895	. . . . 5 |- (A = if(A e. _V, A, (/)) -> (F = {<.A, B>.} <-> F = {<.if(A e. _V, A, (/)), B>.}))
9	5, 8	bibi12d 691	. . . 4 |- (A = if(A e. _V, A, (/)) -> (((F Fn {A} /\ (F` A) = B) <-> F = {<.A, B>.}) <-> ((F Fn {if(A e. _V, A, (/))} /\ (F` if(A e. _V, A, (/))) = B) <-> F = {<.if(A e. _V, A, (/)), B>.})))
10	9	imbi2d 674	. . 3 |- (A = if(A e. _V, A, (/)) -> ((B e. _V -> ((F Fn {A} /\ (F` A) = B) <-> F = {<.A, B>.})) <-> (B e. _V -> ((F Fn {if(A e. _V, A, (/))} /\ (F` if(A e. _V, A, (/))) = B) <-> F = {<.if(A e. _V, A, (/)), B>.}))))
11		0ex 3446	. . . . 5 |- (/) e. _V
12	11	elimel 3025	. . . 4 |- if(A e. _V, A, (/)) e. _V
13	12	bnj147 12480	. . 3 |- (B e. _V -> ((F Fn {if(A e. _V, A, (/))} /\ (F` if(A e. _V, A, (/))) = B) <-> F = {<.if(A e. _V, A, (/)), B>.}))
14	10, 13	dedth 3011	. 2 |- (A e. _V -> (B e. _V -> ((F Fn {A} /\ (F` A) = B) <-> F = {<.A, B>.})))
15	14	imp 377	1 |- ((A e. _V /\ B e. _V) -> ((F Fn {A} /\ (F` A) = B) <-> F = {<.A, B>.}))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  ifcif 2982  {csn 3044  <.cop 3046   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  bnj149 13241

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-if 2983  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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