Theorem 11st22nd 10458

Description: A theorem of the 1st2nd 4108 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 4108	. . 3 |- ((Rel B /\ A e. B) -> A = <.(1st` A), (2nd` A)>.)
2	1	3ad2antl1 809	. 2 |- (((Rel B /\ Rel dom B /\ Rel ran B) /\ A e. B) -> A = <.(1st` A), (2nd` A)>.)
3		1st2nd 4108	. . . . . . . 8 |- ((Rel dom B /\ (1st` A) e. dom B) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
4		1stdm 4109	. . . . . . . 8 |- ((Rel B /\ A e. B) -> (1st` A) e. dom B)
5	3, 4	sylan2 451	. . . . . . 7 |- ((Rel dom B /\ (Rel B /\ A e. B)) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
6	5	exp32 377	. . . . . 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 33	. . . 4 |- (Rel B -> (Rel dom B -> (Rel ran B -> (A e. B -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.))))
9	8	3imp1 846	. . 3 |- (((Rel B /\ Rel dom B /\ Rel ran B) /\ A e. B) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
10		1st2nd 4108	. . . . . . . 8 |- ((Rel ran B /\ (2nd` A) e. ran B) -> (2nd` A) = <.(1st` (2nd` A)), (2nd` (2nd` A))>.)
11		2ndrn 4110	. . . . . . . 8 |- ((Rel B /\ A e. B) -> (2nd` A) e. ran B)
12	10, 11	sylan2 451	. . . . . . 7 |- ((Rel ran B /\ (Rel B /\ A e. B)) -> (2nd` A) = <.(1st` (2nd` A)), (2nd` (2nd` A))>.)
13	12	exp32 377	. . . . . 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 35	. . . 4 |- (Rel B -> (Rel dom B -> (Rel ran B -> (A e. B -> (2nd` A) = <.(1st` (2nd` A)), (2nd` (2nd` A))>.))))
16	15	3imp1 846	. . 3 |- (((Rel B /\ Rel dom B /\ Rel ran B) /\ A e. B) -> (2nd` A) = <.(1st` (2nd` A)), (2nd` (2nd` A))>.)
17	9, 16	opeq12d 2495	. 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 1507	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 223   /\ w3a 775   = wceq 956   e. wcel 958  <.cop 2411  dom cdm 3170  ran crn 3171  Rel wrel 3175  ` cfv 3182  1stc1st 4077  2ndc2nd 4078

This theorem is referenced by:  dedalg 10676  catded 10697

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-1st 4079  df-2nd 4080

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