Theorem bnj147 12480

Description: First-order logic and set theory.

Hypothesis

Ref	Expression
bnj147.1	|- A e. _V

Assertion

Ref	Expression
bnj147	|- (B e. _V -> ((F Fn {A} /\ (F` A) = B) <-> F = {<.A, B>.}))

Proof of Theorem bnj147

Step	Hyp	Ref	Expression
1		eqeq2 1893	. . . 4 |- (B = if(B e. _V, B, (/)) -> ((F` A) = B <-> (F` A) = if(B e. _V, B, (/))))
2	1	anbi2d 678	. . 3 |- (B = if(B e. _V, B, (/)) -> ((F Fn {A} /\ (F` A) = B) <-> (F Fn {A} /\ (F` A) = if(B e. _V, B, (/)))))
3		opeq2 3159	. . . . 5 |- (B = if(B e. _V, B, (/)) -> <.A, B>. = <.A, if(B e. _V, B, (/))>.)
4	3	sneqd 3056	. . . 4 |- (B = if(B e. _V, B, (/)) -> {<.A, B>.} = {<.A, if(B e. _V, B, (/))>.})
5	4	eqeq2d 1895	. . 3 |- (B = if(B e. _V, B, (/)) -> (F = {<.A, B>.} <-> F = {<.A, if(B e. _V, B, (/))>.}))
6	2, 5	bibi12d 691	. 2 |- (B = if(B e. _V, B, (/)) -> (((F Fn {A} /\ (F` A) = B) <-> F = {<.A, B>.}) <-> ((F Fn {A} /\ (F` A) = if(B e. _V, B, (/))) <-> F = {<.A, if(B e. _V, B, (/))>.})))
7		bnj147.1	. . 3 |- A e. _V
8		0ex 3446	. . . 4 |- (/) e. _V
9	8	elimel 3025	. . 3 |- if(B e. _V, B, (/)) e. _V
10	7, 9	bnj134 12478	. 2 |- ((F Fn {A} /\ (F` A) = if(B e. _V, B, (/))) <-> F = {<.A, if(B e. _V, B, (/))>.})
11	6, 10	dedth 3011	1 |- (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:  bnj148 12481  bnj583 13296

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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