2ndval2 - Metamath Proof Explorer

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Theorem 2ndval2 7949

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 5698	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 3442	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3442	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op2nd 7940	. . . . 5 ⊢ (2^nd ‘⟨𝑥, 𝑦⟩) = 𝑦
5	2, 3	op2ndb 6180	. . . . 5 ⊢ ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩} = 𝑦
6	4, 5	eqtr4i 2755	. . . 4 ⊢ (2^nd ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩}
7		fveq2 6826	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = (2^nd ‘⟨𝑥, 𝑦⟩))
8		sneq 4589	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
9	8	cnveqd 5822	. . . . . . 7 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ^◡{𝐴} = ^◡{⟨𝑥, 𝑦⟩})
10	9	inteqd 4904	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ^◡{𝐴} = ∩ ^◡{⟨𝑥, 𝑦⟩})
11	10	inteqd 4904	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ^◡{𝐴} = ∩ ∩ ^◡{⟨𝑥, 𝑦⟩})
12	11	inteqd 4904	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ∩ ^◡{𝐴} = ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩})
13	6, 7, 12	3eqtr4a 2790	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = ∩ ∩ ∩ ^◡{𝐴})
14	13	exlimivv 1932	. 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 1540 ∃wex 1779 ∈ wcel 2109 Vcvv 3438 {csn 4579 ⟨cop 4585 ∩ cint 4899 × cxp 5621 ^◡ccnv 5622 ‘cfv 6486 2^nd c2nd 7930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pr 5374 ax-un 7675

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-iota 6442 df-fun 6488 df-fv 6494 df-2nd 7932

This theorem is referenced by: (None)

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