Theorem 2ndval2 6223

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 4726	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 2766	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2766	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op2nd 6214	. . . . 5 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
5	2, 3	op2ndb 5154	. . . . 5 ⊢ ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩} = 𝑦
6	4, 5	eqtr4i 2220	. . . 4 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩}
7		fveq2 5561	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = (2nd ‘⟨𝑥, 𝑦⟩))
8		sneq 3634	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
9	8	cnveqd 4843	. . . . . . 7 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ◡{𝐴} = ◡{⟨𝑥, 𝑦⟩})
10	9	inteqd 3880	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ◡{𝐴} = ∩ ◡{⟨𝑥, 𝑦⟩})
11	10	inteqd 3880	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ◡{𝐴} = ∩ ∩ ◡{⟨𝑥, 𝑦⟩})
12	11	inteqd 3880	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ∩ ◡{𝐴} = ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩})
13	6, 7, 12	3eqtr4a 2255	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴})
14	13	exlimivv 1911	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴})
15	1, 14	sylbi 121	1 ⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴})

Syntax hints:   → wi 4   = wceq 1364  ∃wex 1506   ∈ wcel 2167  Vcvv 2763  {csn 3623  ⟨cop 3626  ∩ cint 3875   × cxp 4662  ^◡ccnv 4663  ‘cfv 5259  2^nd c2nd 6206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fv 5267  df-2nd 6208

This theorem is referenced by: (None)