Theorem 2ndval2 6062

Description: Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)

Assertion

Ref	Expression
2ndval2	⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴})

Proof of Theorem 2ndval2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elvv 4609	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 2692	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2692	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op2nd 6053	. . . . 5 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
5	2, 3	op2ndb 5030	. . . . 5 ⊢ ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩} = 𝑦
6	4, 5	eqtr4i 2164	. . . 4 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩}
7		fveq2 5429	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = (2nd ‘⟨𝑥, 𝑦⟩))
8		sneq 3543	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
9	8	cnveqd 4723	. . . . . . 7 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ◡{𝐴} = ◡{⟨𝑥, 𝑦⟩})
10	9	inteqd 3784	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ◡{𝐴} = ∩ ◡{⟨𝑥, 𝑦⟩})
11	10	inteqd 3784	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ◡{𝐴} = ∩ ∩ ◡{⟨𝑥, 𝑦⟩})
12	11	inteqd 3784	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ∩ ◡{𝐴} = ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩})
13	6, 7, 12	3eqtr4a 2199	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴})
14	13	exlimivv 1869	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴})
15	1, 14	sylbi 120	1 ⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴})

Syntax hints:   → wi 4   = wceq 1332  ∃wex 1469   ∈ wcel 1481  Vcvv 2689  {csn 3532  ⟨cop 3535  ∩ cint 3779   × cxp 4545  ^◡ccnv 4546  ‘cfv 5131  2^nd c2nd 6045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fv 5139  df-2nd 6047

This theorem is referenced by: (None)