Theorem 2ndval2 6308

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 4781	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 2802	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2802	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op2nd 6299	. . . . 5 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
5	2, 3	op2ndb 5212	. . . . 5 ⊢ ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩} = 𝑦
6	4, 5	eqtr4i 2253	. . . 4 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩}
7		fveq2 5629	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = (2nd ‘⟨𝑥, 𝑦⟩))
8		sneq 3677	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
9	8	cnveqd 4898	. . . . . . 7 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ◡{𝐴} = ◡{⟨𝑥, 𝑦⟩})
10	9	inteqd 3928	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ◡{𝐴} = ∩ ◡{⟨𝑥, 𝑦⟩})
11	10	inteqd 3928	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ◡{𝐴} = ∩ ∩ ◡{⟨𝑥, 𝑦⟩})
12	11	inteqd 3928	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ∩ ◡{𝐴} = ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩})
13	6, 7, 12	3eqtr4a 2288	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴})
14	13	exlimivv 1943	. 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 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799  {csn 3666  ⟨cop 3669  ∩ cint 3923   × cxp 4717  ^◡ccnv 4718  ‘cfv 5318  2^nd c2nd 6291

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fv 5326  df-2nd 6293

This theorem is referenced by: (None)