Theorem 2ndval2 6324

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 4790	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 2804	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2804	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op2nd 6315	. . . . 5 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
5	2, 3	op2ndb 5222	. . . . 5 ⊢ ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩} = 𝑦
6	4, 5	eqtr4i 2254	. . . 4 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩}
7		fveq2 5642	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = (2nd ‘⟨𝑥, 𝑦⟩))
8		sneq 3681	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
9	8	cnveqd 4908	. . . . . . 7 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ◡{𝐴} = ◡{⟨𝑥, 𝑦⟩})
10	9	inteqd 3934	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ◡{𝐴} = ∩ ◡{⟨𝑥, 𝑦⟩})
11	10	inteqd 3934	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ◡{𝐴} = ∩ ∩ ◡{⟨𝑥, 𝑦⟩})
12	11	inteqd 3934	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ∩ ◡{𝐴} = ∩ ∩ ∩ ◡{⟨𝑥, 𝑦⟩})
13	6, 7, 12	3eqtr4a 2289	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴})
14	13	exlimivv 1944	. 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 1397  ∃wex 1540   ∈ wcel 2201  Vcvv 2801  {csn 3670  ⟨cop 3673  ∩ cint 3929   × cxp 4725  ^◡ccnv 4726  ‘cfv 5328  2^nd c2nd 6307

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-iota 5288  df-fun 5330  df-fv 5336  df-2nd 6309

This theorem is referenced by: (None)