el2xptp0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > el2xptp0		Structured version Visualization version GIF version

Theorem el2xptp0 7978

Description: A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)

**Assertion**
Ref	Expression
el2xptp0	⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍)) ↔ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩))

**Proof of Theorem el2xptp0**
Step	Hyp	Ref	Expression
1		xp1st 7963	. . . . . 6 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (1^st ‘𝐴) ∈ (𝑈 × 𝑉))
2	1	ad2antrl 728	. . . . 5 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → (1^st ‘𝐴) ∈ (𝑈 × 𝑉))
3		3simpa 1148	. . . . . . 7 ⊢ (((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍) → ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌))
4	3	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍)) → ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌))
5	4	adantl 481	. . . . 5 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌))
6		eqopi 7967	. . . . 5 ⊢ (((1^st ‘𝐴) ∈ (𝑈 × 𝑉) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌)) → (1^st ‘𝐴) = ⟨𝑋, 𝑌⟩)
7	2, 5, 6	syl2anc 584	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → (1^st ‘𝐴) = ⟨𝑋, 𝑌⟩)
8		simprr3 1224	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → (2^nd ‘𝐴) = 𝑍)
9	7, 8	jca 511	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍))
10		df-ot 4587	. . . . . 6 ⊢ ⟨𝑋, 𝑌, 𝑍⟩ = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩
11	10	eqeq2i 2747	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ 𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩)
12		eqop 7973	. . . . 5 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ↔ ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍)))
13	11, 12	bitrid 283	. . . 4 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍)))
14	13	ad2antrl 728	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍)))
15	9, 14	mpbird 257	. 2 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩)
16		opelxpi 5659	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
17	16	3adant3 1132	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
18		simp3 1138	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ 𝑊)
19	17, 18	opelxpd 5661	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
20	10, 19	eqeltrid 2838	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
21	20	adantr 480	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
22		eleq1 2822	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
23	22	adantl 481	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
24	21, 23	mpbird 257	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → 𝐴 ∈ ((𝑈 × 𝑉) × 𝑊))
25		2fveq3 6837	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (1^st ‘(1^st ‘𝐴)) = (1^st ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)))
26		ot1stg 7945	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (1^st ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑋)
27	25, 26	sylan9eqr 2791	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (1^st ‘(1^st ‘𝐴)) = 𝑋)
28		2fveq3 6837	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2^nd ‘(1^st ‘𝐴)) = (2^nd ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)))
29		ot2ndg 7946	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (2^nd ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑌)
30	28, 29	sylan9eqr 2791	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2^nd ‘(1^st ‘𝐴)) = 𝑌)
31		fveq2 6832	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2^nd ‘𝐴) = (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩))
32		ot3rdg 7947	. . . . . 6 ⊢ (𝑍 ∈ 𝑊 → (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
33	32	3ad2ant3 1135	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
34	31, 33	sylan9eqr 2791	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2^nd ‘𝐴) = 𝑍)
35	27, 30, 34	3jca 1128	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))
36	24, 35	jca 511	. 2 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍)))
37	15, 36	impbida 800	1 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍)) ↔ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ⟨cop 4584 ⟨cotp 4586 × cxp 5620 ‘cfv 6490 1^st c1st 7929 2^nd c2nd 7930

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 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pr 5375 ax-un 7678

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-sn 4579 df-pr 4581 df-op 4585 df-ot 4587 df-uni 4862 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-iota 6446 df-fun 6492 df-fv 6498 df-1st 7931 df-2nd 7932

This theorem is referenced by: (None)

W3C validator