Theorem bnj99 12450

Description: First-order logic and set theory.

Hypothesis

Ref	Expression
bnj99.1	|- Y e. _V

Assertion

Ref	Expression
bnj99	|- ([Y / x]x Fn Z <-> Y Fn Z)

Distinct variable group:   x,Z

Proof of Theorem bnj99

Step	Hyp	Ref	Expression
1		bnj99.1	. . 3 |- Y e. _V
2		sbc8g 2477	. . 3 |- (Y e. _V -> ([Y / x]x Fn Z <-> Y e. {x | x Fn Z}))
3	1, 2	ax-mp 7	. 2 |- ([Y / x]x Fn Z <-> Y e. {x | x Fn Z})
4	1	isseti 2297	. . . . 5 |- E.w w = Y
5		fneq1 4503	. . . . . 6 |- (w = Y -> (w Fn Z <-> Y Fn Z))
6		eleq1 1957	. . . . . . 7 |- (w = Y -> (w e. {w | w Fn Z} <-> Y e. {w | w Fn Z}))
7		abid 1873	. . . . . . 7 |- (w e. {w | w Fn Z} <-> w Fn Z)
8	6, 7	syl5bbr 593	. . . . . 6 |- (w = Y -> (w Fn Z <-> Y e. {w | w Fn Z}))
9	5, 8	bitr3d 589	. . . . 5 |- (w = Y -> (Y Fn Z <-> Y e. {w | w Fn Z}))
10	4, 9	bnj101 12448	. . . 4 |- E.w(Y Fn Z <-> Y e. {w | w Fn Z})
11		fneq1 4503	. . . . . . . 8 |- (x = w -> (x Fn Z <-> w Fn Z))
12	11	cbvabv 2420	. . . . . . 7 |- {x | x Fn Z} = {w | w Fn Z}
13	12	eleq2i 1961	. . . . . 6 |- (Y e. {x | x Fn Z} <-> Y e. {w | w Fn Z})
14	13	bibi2i 669	. . . . 5 |- ((Y Fn Z <-> Y e. {x | x Fn Z}) <-> (Y Fn Z <-> Y e. {w | w Fn Z}))
15	14	exbii 1398	. . . 4 |- (E.w(Y Fn Z <-> Y e. {x | x Fn Z}) <-> E.w(Y Fn Z <-> Y e. {w | w Fn Z}))
16	10, 15	mpbir 207	. . 3 |- E.w(Y Fn Z <-> Y e. {x | x Fn Z})
17		ax-17 1317	. . . 4 |- ((Y Fn Z <-> Y e. {x | x Fn Z}) -> A.w(Y Fn Z <-> Y e. {x | x Fn Z}))
18	17	19.9 1383	. . 3 |- (E.w(Y Fn Z <-> Y e. {x | x Fn Z}) <-> (Y Fn Z <-> Y e. {x | x Fn Z}))
19	16, 18	mpbi 206	. 2 |- (Y Fn Z <-> Y e. {x | x Fn Z})
20	3, 19	bitr4i 193	1 |- ([Y / x]x Fn Z <-> Y Fn Z)

Syntax hints:   <-> wb 163   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  {cab 1871  _Vcvv 2292   Fn wfn 3993

This theorem is referenced by:  bnj211OLD 12505  bnj109 13226  bnj124 13234

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-sbc 2454  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-id 3586  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-fun 4008  df-fn 4009

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