Theorem 2ndval2 6349

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 4811	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 2815	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2815	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op2nd 6340	. . . . 5 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
5	2, 3	op2ndb 5245	. . . . 5 ⊢ ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩} = 𝑦
6	4, 5	eqtr4i 2256	. . . 4 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩}
7		fveq2 5669	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = (2nd ‘⟨𝑥, 𝑦⟩))
8		sneq 3699	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
9	8	cnveqd 4930	. . . . . . 7 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ◡{𝐴} = ◡{⟨𝑥, 𝑦⟩})
10	9	inteqd 3953	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ◡{𝐴} = ∩ ◡{⟨𝑥, 𝑦⟩})
11	10	inteqd 3953	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ◡{𝐴} = ∩ ∩ ◡{⟨𝑥, 𝑦⟩})
12	11	inteqd 3953	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ∩ ◡{𝐴} = ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩})
13	6, 7, 12	3eqtr4a 2291	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴})
14	13	exlimivv 1946	. 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 1398  ∃wex 1541   ∈ wcel 2203  Vcvv 2812  {csn 3688  ⟨cop 3691  ∩ cint 3948   × cxp 4746  ^◡ccnv 4747  ‘cfv 5351  2^nd c2nd 6332

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-iota 5311  df-fun 5353  df-fv 5359  df-2nd 6334

This theorem is referenced by: (None)