2ndval2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 2ndval2		Structured version Visualization version GIF version

Theorem 2ndval2 7951

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) → (2^nd ‘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 5699	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 3444	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3444	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op2nd 7942	. . . . 5 ⊢ (2^nd ‘⟨𝑥, 𝑦⟩) = 𝑦
5	2, 3	op2ndb 6185	. . . . 5 ⊢ ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩} = 𝑦
6	4, 5	eqtr4i 2762	. . . 4 ⊢ (2^nd ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩}
7		fveq2 6834	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = (2^nd ‘⟨𝑥, 𝑦⟩))
8		sneq 4590	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
9	8	cnveqd 5824	. . . . . . 7 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ^◡{𝐴} = ^◡{⟨𝑥, 𝑦⟩})
10	9	inteqd 4907	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ^◡{𝐴} = ∩ ^◡{⟨𝑥, 𝑦⟩})
11	10	inteqd 4907	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ^◡{𝐴} = ∩ ∩ ^◡{⟨𝑥, 𝑦⟩})
12	11	inteqd 4907	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ∩ ^◡{𝐴} = ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩})
13	6, 7, 12	3eqtr4a 2797	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = ∩ ∩ ∩ ^◡{𝐴})
14	13	exlimivv 1933	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = ∩ ∩ ∩ ^◡{𝐴})
15	1, 14	sylbi 217	1 ⊢ (𝐴 ∈ (V × V) → (2^nd ‘𝐴) = ∩ ∩ ∩ ^◡{𝐴})

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∃wex 1780 ∈ wcel 2113 Vcvv 3440 {csn 4580 ⟨cop 4586 ∩ cint 4902 × cxp 5622 ^◡ccnv 5623 ‘cfv 6492 2^nd c2nd 7932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-iota 6448 df-fun 6494 df-fv 6500 df-2nd 7934

This theorem is referenced by: (None)

W3C validator