Theorem btmp 15252

Description: Belonging to (tarskiMap` A).

Assertion

Ref	Expression
btmp	|- ((A e. C /\ B e. D) -> (B e. (tarskiMap` A) <-> A.x e. Tarski (A e. x -> B e. x)))

Distinct variable groups:   x,A   x,B

Proof of Theorem btmp

Step	Hyp	Ref	Expression
1		tarval2g 15250	. . . 4 |- (A e. C -> (tarskiMap` A) = |^|{x | (A e. x /\ x e. Tarski )})
2	1	adantr 425	. . 3 |- ((A e. C /\ B e. D) -> (tarskiMap` A) = |^|{x | (A e. x /\ x e. Tarski )})
3	2	eleq2d 1964	. 2 |- ((A e. C /\ B e. D) -> (B e. (tarskiMap` A) <-> B e. |^|{x | (A e. x /\ x e. Tarski )}))
4		elintg 3222	. . 3 |- (B e. D -> (B e. |^|{x | (A e. x /\ x e. Tarski )} <-> A.y e. {x | (A e. x /\ x e. Tarski )}B e. y))
5	4	adantl 424	. 2 |- ((A e. C /\ B e. D) -> (B e. |^|{x | (A e. x /\ x e. Tarski )} <-> A.y e. {x | (A e. x /\ x e. Tarski )}B e. y))
6		ax-17 1317	. . . . . . 7 |- ((A e. C /\ B e. D) -> A.y(A e. C /\ B e. D))
7		pm2.27 76	. . . . . . . . . . 11 |- ((A e. y /\ y e. Tarski ) -> (((A e. y /\ y e. Tarski ) -> B e. y) -> B e. y))
8	7	ex 402	. . . . . . . . . 10 |- (A e. y -> (y e. Tarski -> (((A e. y /\ y e. Tarski ) -> B e. y) -> B e. y)))
9	8	com13 37	. . . . . . . . 9 |- (((A e. y /\ y e. Tarski ) -> B e. y) -> (y e. Tarski -> (A e. y -> B e. y)))
10		pm2.27 76	. . . . . . . . . . . 12 |- (y e. Tarski -> ((y e. Tarski -> (A e. y -> B e. y)) -> (A e. y -> B e. y)))
11	10	com3r 39	. . . . . . . . . . 11 |- (A e. y -> (y e. Tarski -> ((y e. Tarski -> (A e. y -> B e. y)) -> B e. y)))
12	11	imp 377	. . . . . . . . . 10 |- ((A e. y /\ y e. Tarski ) -> ((y e. Tarski -> (A e. y -> B e. y)) -> B e. y))
13	12	com12 14	. . . . . . . . 9 |- ((y e. Tarski -> (A e. y -> B e. y)) -> ((A e. y /\ y e. Tarski ) -> B e. y))
14	9, 13	impbii 174	. . . . . . . 8 |- (((A e. y /\ y e. Tarski ) -> B e. y) <-> (y e. Tarski -> (A e. y -> B e. y)))
15	14	a1i 8	. . . . . . 7 |- ((A e. C /\ B e. D) -> (((A e. y /\ y e. Tarski ) -> B e. y) <-> (y e. Tarski -> (A e. y -> B e. y))))
16	6, 15	albid 1459	. . . . . 6 |- ((A e. C /\ B e. D) -> (A.y((A e. y /\ y e. Tarski ) -> B e. y) <-> A.y(y e. Tarski -> (A e. y -> B e. y))))
17		df-ral 2109	. . . . . 6 |- (A.y e. Tarski (A e. y -> B e. y) <-> A.y(y e. Tarski -> (A e. y -> B e. y)))
18	16, 17	syl6bbr 597	. . . . 5 |- ((A e. C /\ B e. D) -> (A.y((A e. y /\ y e. Tarski ) -> B e. y) <-> A.y e. Tarski (A e. y -> B e. y)))
19		eleq2 1958	. . . . . . 7 |- (y = x -> (A e. y <-> A e. x))
20		eleq2 1958	. . . . . . 7 |- (y = x -> (B e. y <-> B e. x))
21	19, 20	imbi12d 688	. . . . . 6 |- (y = x -> ((A e. y -> B e. y) <-> (A e. x -> B e. x)))
22	21	cbvralv 2280	. . . . 5 |- (A.y e. Tarski (A e. y -> B e. y) <-> A.x e. Tarski (A e. x -> B e. x))
23	18, 22	syl6bb 595	. . . 4 |- ((A e. C /\ B e. D) -> (A.y((A e. y /\ y e. Tarski ) -> B e. y) <-> A.x e. Tarski (A e. x -> B e. x)))
24		visset 2295	. . . . . . 7 |- y e. _V
25		eleq2 1958	. . . . . . . . 9 |- (x = y -> (A e. x <-> A e. y))
26		eleq1 1957	. . . . . . . . 9 |- (x = y -> (x e. Tarski <-> y e. Tarski ))
27	25, 26	anbi12d 690	. . . . . . . 8 |- (x = y -> ((A e. x /\ x e. Tarski ) <-> (A e. y /\ y e. Tarski )))
28	27	elabg 2405	. . . . . . 7 |- (y e. _V -> (y e. {x | (A e. x /\ x e. Tarski )} <-> (A e. y /\ y e. Tarski )))
29	24, 28	ax-mp 7	. . . . . 6 |- (y e. {x | (A e. x /\ x e. Tarski )} <-> (A e. y /\ y e. Tarski ))
30	29	imbi1i 203	. . . . 5 |- ((y e. {x | (A e. x /\ x e. Tarski )} -> B e. y) <-> ((A e. y /\ y e. Tarski ) -> B e. y))
31	30	albii 1346	. . . 4 |- (A.y(y e. {x | (A e. x /\ x e. Tarski )} -> B e. y) <-> A.y((A e. y /\ y e. Tarski ) -> B e. y))
32	23, 31	syl5bb 591	. . 3 |- ((A e. C /\ B e. D) -> (A.y(y e. {x | (A e. x /\ x e. Tarski )} -> B e. y) <-> A.x e. Tarski (A e. x -> B e. x)))
33		df-ral 2109	. . 3 |- (A.y e. {x | (A e. x /\ x e. Tarski )}B e. y <-> A.y(y e. {x | (A e. x /\ x e. Tarski )} -> B e. y))
34	32, 33	syl5bb 591	. 2 |- ((A e. C /\ B e. D) -> (A.y e. {x | (A e. x /\ x e. Tarski )}B e. y <-> A.x e. Tarski (A e. x -> B e. x)))
35	3, 5, 34	3bitrd 603	1 |- ((A e. C /\ B e. D) -> (B e. (tarskiMap` A) <-> A.x e. Tarski (A e. x -> B e. x)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  _Vcvv 2292  |^|cint 3214  ` cfv 3998   Tarski ctarski 15208  tarskiMapctarskim 15209

This theorem is referenced by:  pwtsm 15266  subtsm 15267

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-groth 10131

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-v 2294  df-sbc 2454  df-csb 2541  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-int 3215  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-fv 4014  df-tsk 15210  df-tskmp 15248

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