Theorem 11st22nd 14348

Description: A theorem of the 1st2nd 5048 family.

Assertion

Ref	Expression
11st22nd	|- (((Rel B /\ Rel dom B /\ Rel ran B) /\ A e. B) -> A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>.)

Proof of Theorem 11st22nd

Step	Hyp	Ref	Expression
1		1st2nd 5048	. . 3 |- ((Rel B /\ A e. B) -> A = <.(1st` A), (2nd` A)>.)
2	1	3ad2antl1 1038	. 2 |- (((Rel B /\ Rel dom B /\ Rel ran B) /\ A e. B) -> A = <.(1st` A), (2nd` A)>.)
3		1st2nd 5048	. . . . . . . 8 |- ((Rel dom B /\ (1st` A) e. dom B) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
4		1stdm 5049	. . . . . . . 8 |- ((Rel B /\ A e. B) -> (1st` A) e. dom B)
5	3, 4	sylan2 500	. . . . . . 7 |- ((Rel dom B /\ (Rel B /\ A e. B)) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
6	5	exp32 408	. . . . . 6 |- (Rel dom B -> (Rel B -> (A e. B -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)))
7	6	a1i 8	. . . . 5 |- (Rel ran B -> (Rel dom B -> (Rel B -> (A e. B -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.))))
8	7	com13 37	. . . 4 |- (Rel B -> (Rel dom B -> (Rel ran B -> (A e. B -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.))))
9	8	3imp1 1081	. . 3 |- (((Rel B /\ Rel dom B /\ Rel ran B) /\ A e. B) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
10		1st2nd 5048	. . . . . . . 8 |- ((Rel ran B /\ (2nd` A) e. ran B) -> (2nd` A) = <.(1st` (2nd` A)), (2nd` (2nd` A))>.)
11		2ndrn 5050	. . . . . . . 8 |- ((Rel B /\ A e. B) -> (2nd` A) e. ran B)
12	10, 11	sylan2 500	. . . . . . 7 |- ((Rel ran B /\ (Rel B /\ A e. B)) -> (2nd` A) = <.(1st` (2nd` A)), (2nd` (2nd` A))>.)
13	12	exp32 408	. . . . . 6 |- (Rel ran B -> (Rel B -> (A e. B -> (2nd` A) = <.(1st` (2nd` A)), (2nd` (2nd` A))>.)))
14	13	a1i 8	. . . . 5 |- (Rel dom B -> (Rel ran B -> (Rel B -> (A e. B -> (2nd` A) = <.(1st` (2nd` A)), (2nd` (2nd` A))>.))))
15	14	com3r 39	. . . 4 |- (Rel B -> (Rel dom B -> (Rel ran B -> (A e. B -> (2nd` A) = <.(1st` (2nd` A)), (2nd` (2nd` A))>.))))
16	15	3imp1 1081	. . 3 |- (((Rel B /\ Rel dom B /\ Rel ran B) /\ A e. B) -> (2nd` A) = <.(1st` (2nd` A)), (2nd` (2nd` A))>.)
17	9, 16	opeq12d 3166	. 2 |- (((Rel B /\ Rel dom B /\ Rel ran B) /\ A e. B) -> <.(1st` A), (2nd` A)>. = <.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>.)
18	2, 17	eqtrd 1925	1 |- (((Rel B /\ Rel dom B /\ Rel ran B) /\ A e. B) -> A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>.)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  <.cop 3046  dom cdm 3986  ran crn 3987  Rel wrel 3991  ` cfv 3998  1stc1st 5018  2ndc2nd 5019

This theorem is referenced by:  dedalg 15090  catded 15111

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-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-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-1st 5020  df-2nd 5021

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