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 7710

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 5629	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 3500	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3500	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op2nd 7701	. . . . 5 ⊢ (2^nd ‘⟨𝑥, 𝑦⟩) = 𝑦
5	2, 3	op2ndb 6087	. . . . 5 ⊢ ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩} = 𝑦
6	4, 5	eqtr4i 2850	. . . 4 ⊢ (2^nd ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩}
7		fveq2 6673	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = (2^nd ‘⟨𝑥, 𝑦⟩))
8		sneq 4580	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
9	8	cnveqd 5749	. . . . . . 7 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ^◡{𝐴} = ^◡{⟨𝑥, 𝑦⟩})
10	9	inteqd 4884	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ^◡{𝐴} = ∩ ^◡{⟨𝑥, 𝑦⟩})
11	10	inteqd 4884	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ^◡{𝐴} = ∩ ∩ ^◡{⟨𝑥, 𝑦⟩})
12	11	inteqd 4884	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ ∩ ^◡{𝐴} = ∩ ∩ ∩ ^◡{⟨𝑥, 𝑦⟩})
13	6, 7, 12	3eqtr4a 2885	. . 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 219	1 ⊢ (𝐴 ∈ (V × V) → (2^nd ‘𝐴) = ∩ ∩ ∩ ^◡{𝐴})

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1536 ∃wex 1779 ∈ wcel 2113 Vcvv 3497 {csn 4570 ⟨cop 4576 ∩ cint 4879 × cxp 5556 ^◡ccnv 5557 ‘cfv 6358 2^nd c2nd 7691

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-int 4880 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fv 6366 df-2nd 7693

This theorem is referenced by: (None)

W3C validator