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Theorem el2xptp0 7103

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 7089	. . . . . 6 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (1^st ‘𝐴) ∈ (𝑈 × 𝑉))
2	1	ad2antrl 760	. . . . 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 1051	. . . . . . 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 7093	. . . . 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 691	. . . 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 1104	. . . 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 553	. . 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 4134	. . . . . 6 ⊢ ⟨𝑋, 𝑌, 𝑍⟩ = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩
11	10	eqeq2i 2622	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ 𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩)
12		eqop 7099	. . . . 5 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ↔ ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍)))
13	11, 12	syl5bb 271	. . . 4 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍)))
14	13	ad2antrl 760	. . 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 246	. 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 5072	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
17	16	3adant3 1074	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
18		simp3 1056	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ 𝑊)
19		opelxp 5070	. . . . . . 7 ⊢ (⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊) ↔ (⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉) ∧ 𝑍 ∈ 𝑊))
20	17, 18, 19	sylanbrc 695	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
21	10, 20	syl5eqel 2692	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
22	21	adantr 480	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
23		eleq1 2676	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
24	23	adantl 481	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
25	22, 24	mpbird 246	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → 𝐴 ∈ ((𝑈 × 𝑉) × 𝑊))
26		fveq2 6103	. . . . . 6 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (1^st ‘𝐴) = (1^st ‘⟨𝑋, 𝑌, 𝑍⟩))
27	26	fveq2d 6107	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (1^st ‘(1^st ‘𝐴)) = (1^st ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)))
28		ot1stg 7073	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (1^st ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑋)
29	27, 28	sylan9eqr 2666	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (1^st ‘(1^st ‘𝐴)) = 𝑋)
30	26	fveq2d 6107	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2^nd ‘(1^st ‘𝐴)) = (2^nd ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)))
31		ot2ndg 7074	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (2^nd ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑌)
32	30, 31	sylan9eqr 2666	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2^nd ‘(1^st ‘𝐴)) = 𝑌)
33		fveq2 6103	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2^nd ‘𝐴) = (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩))
34		ot3rdg 7075	. . . . . 6 ⊢ (𝑍 ∈ 𝑊 → (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
35	34	3ad2ant3 1077	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
36	33, 35	sylan9eqr 2666	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2^nd ‘𝐴) = 𝑍)
37	29, 32, 36	3jca 1235	. . 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))
38	25, 37	jca 553	. 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)))
39	15, 38	impbida 873	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⟨cop 4131 ⟨cotp 4133 × cxp 5036 ‘cfv 5804 1^st c1st 7057 2^nd c2nd 7058

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-ot 4134 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-2nd 7060

This theorem is referenced by: el2wlkonot 26396 el2spthonot 26397

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