2ndval2 - Metamath Proof Explorer

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Theorem 2ndval2 8046

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 5731	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 3458	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3458	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op2nd 8037	. . . . 5 ⊢ (2^nd ‘⟨𝑥, 𝑦⟩) = 𝑦
5	2, 3	op2ndb 6242	. . . . 5 ⊢ ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩} = 𝑦
6	4, 5	eqtr4i 2789	. . . 4 ⊢ (2^nd ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩}
7		fveq2 6911	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = (2^nd ‘⟨𝑥, 𝑦⟩))
8		sneq 4594	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
9	8	cnveqd 5861	. . . . . . 7 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ^◡{𝐴} = ^◡{⟨𝑥, 𝑦⟩})
10	9	inteqd 4915	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ^◡{𝐴} = ∩ ^◡{⟨𝑥, 𝑦⟩})
11	10	inteqd 4915	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ^◡{𝐴} = ∩ ∩ ^◡{⟨𝑥, 𝑦⟩})
12	11	inteqd 4915	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ∩ ^◡{𝐴} = ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩})
13	6, 7, 12	3eqtr4a 2824	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = ∩ ∩ ∩ ^◡{𝐴})
14	13	exlimivv 1967	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = ∩ ∩ ∩ ^◡{𝐴})
15	1, 14	sylbi 220	1 ⊢ (𝐴 ∈ (V × V) → (2^nd ‘𝐴) = ∩ ∩ ∩ ^◡{𝐴})

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∃wex 1812 ∈ wcel 2147 Vcvv 3454 {csn 4588 ⟨cop 4838 ∩ cint 4910 × cxp 5654 ^◡ccnv 5655 ‘cfv 6561 2^nd c2nd 8027

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pr 5396 ax-un 7778

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-iota 6515 df-fun 6563 df-fv 6569 df-2nd 8029

This theorem is used by: (None)

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