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Theorem el2xptp0 7212

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 7198	. . . . . 6 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (1^st ‘𝐴) ∈ (𝑈 × 𝑉))
2	1	ad2antrl 764	. . . . 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 1058	. . . . . . 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 482	. . . . . 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 482	. . . . 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 7202	. . . . 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 693	. . . 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 1111	. . . 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 554	. . 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 4186	. . . . . 6 ⊢ ⟨𝑋, 𝑌, 𝑍⟩ = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩
11	10	eqeq2i 2634	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ 𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩)
12		eqop 7208	. . . . 5 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ↔ ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍)))
13	11, 12	syl5bb 272	. . . 4 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍)))
14	13	ad2antrl 764	. . 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 247	. 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 5148	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
17	16	3adant3 1081	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
18		simp3 1063	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ 𝑊)
19		opelxp 5146	. . . . . . 7 ⊢ (⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊) ↔ (⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉) ∧ 𝑍 ∈ 𝑊))
20	17, 18, 19	sylanbrc 698	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
21	10, 20	syl5eqel 2705	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
22	21	adantr 481	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
23		eleq1 2689	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
24	23	adantl 482	. . . 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 247	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → 𝐴 ∈ ((𝑈 × 𝑉) × 𝑊))
26		fveq2 6191	. . . . . 6 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (1^st ‘𝐴) = (1^st ‘⟨𝑋, 𝑌, 𝑍⟩))
27	26	fveq2d 6195	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (1^st ‘(1^st ‘𝐴)) = (1^st ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)))
28		ot1stg 7182	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (1^st ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑋)
29	27, 28	sylan9eqr 2678	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (1^st ‘(1^st ‘𝐴)) = 𝑋)
30	26	fveq2d 6195	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2^nd ‘(1^st ‘𝐴)) = (2^nd ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)))
31		ot2ndg 7183	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (2^nd ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑌)
32	30, 31	sylan9eqr 2678	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2^nd ‘(1^st ‘𝐴)) = 𝑌)
33		fveq2 6191	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2^nd ‘𝐴) = (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩))
34		ot3rdg 7184	. . . . . 6 ⊢ (𝑍 ∈ 𝑊 → (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
35	34	3ad2ant3 1084	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
36	33, 35	sylan9eqr 2678	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2^nd ‘𝐴) = 𝑍)
37	29, 32, 36	3jca 1242	. . 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 554	. 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 877	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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⟨cop 4183 ⟨cotp 4185 × cxp 5112 ‘cfv 5888 1^st c1st 7166 2^nd c2nd 7167

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-ot 4186 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-1st 7168 df-2nd 7169

This theorem is referenced by: (None)

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