Theorem op2ndb 4747

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4175 to extract the first member and op2nda 4748 for an alternate version.) (Contributed by NM, 25-Nov-2003.)

Hypotheses

Ref	Expression
cnvsn.1	⊢ A ∈ V
cnvsn.2	⊢ B ∈ V

Assertion

Ref	Expression
op2ndb	⊢ ∩ ∩ ∩ ◡{⟨A, B⟩} = B

Proof of Theorem op2ndb

Step	Hyp	Ref	Expression
1		cnvsn.1	. . . . . . 7 ⊢ A ∈ V
2		cnvsn.2	. . . . . . 7 ⊢ B ∈ V
3	1, 2	cnvsn 4746	. . . . . 6 ⊢ ◡{⟨A, B⟩} = {⟨B, A⟩}
4	3	inteqi 3610	. . . . 5 ⊢ ∩ ◡{⟨A, B⟩} = ∩ {⟨B, A⟩}
5		opexgOLD 3956	. . . . . . 7 ⊢ ((B ∈ V ∧ A ∈ V) → ⟨B, A⟩ ∈ V)
6	2, 1, 5	mp2an 402	. . . . . 6 ⊢ ⟨B, A⟩ ∈ V
7	6	intsn 3641	. . . . 5 ⊢ ∩ {⟨B, A⟩} = ⟨B, A⟩
8	4, 7	eqtri 2057	. . . 4 ⊢ ∩ ◡{⟨A, B⟩} = ⟨B, A⟩
9	8	inteqi 3610	. . 3 ⊢ ∩ ∩ ◡{⟨A, B⟩} = ∩ ⟨B, A⟩
10	9	inteqi 3610	. 2 ⊢ ∩ ∩ ∩ ◡{⟨A, B⟩} = ∩ ∩ ⟨B, A⟩
11	2, 1	op1stb 4175	. 2 ⊢ ∩ ∩ ⟨B, A⟩ = B
12	10, 11	eqtri 2057	1 ⊢ ∩ ∩ ∩ ◡{⟨A, B⟩} = B

Syntax hints:   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {csn 3367  ⟨cop 3370  ∩ cint 3606  ^◡ccnv 4287

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-int 3607  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296

This theorem is referenced by:  2ndval2  5725