2ndval2 - Metamath Proof Explorer

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Theorem 2ndval2 7822

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 5652	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 3426	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3426	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op2nd 7813	. . . . 5 ⊢ (2^nd ‘⟨𝑥, 𝑦⟩) = 𝑦
5	2, 3	op2ndb 6119	. . . . 5 ⊢ ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩} = 𝑦
6	4, 5	eqtr4i 2769	. . . 4 ⊢ (2^nd ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩}
7		fveq2 6756	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = (2^nd ‘⟨𝑥, 𝑦⟩))
8		sneq 4568	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
9	8	cnveqd 5773	. . . . . . 7 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ^◡{𝐴} = ^◡{⟨𝑥, 𝑦⟩})
10	9	inteqd 4881	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ^◡{𝐴} = ∩ ^◡{⟨𝑥, 𝑦⟩})
11	10	inteqd 4881	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ^◡{𝐴} = ∩ ∩ ^◡{⟨𝑥, 𝑦⟩})
12	11	inteqd 4881	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ∩ ^◡{𝐴} = ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩})
13	6, 7, 12	3eqtr4a 2805	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = ∩ ∩ ∩ ^◡{𝐴})
14	13	exlimivv 1936	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = ∩ ∩ ∩ ^◡{𝐴})
15	1, 14	sylbi 216	1 ⊢ (𝐴 ∈ (V × V) → (2^nd ‘𝐴) = ∩ ∩ ∩ ^◡{𝐴})

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∃wex 1783 ∈ wcel 2108 Vcvv 3422 {csn 4558 ⟨cop 4564 ∩ cint 4876 × cxp 5578 ^◡ccnv 5579 ‘cfv 6418 2^nd c2nd 7803

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-int 4877 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-iota 6376 df-fun 6420 df-fv 6426 df-2nd 7805

This theorem is referenced by: (None)

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