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Theorem el2xptp0 8039

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 8024	. . . . . 6 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (1^st ‘𝐴) ∈ (𝑈 × 𝑉))
2	1	ad2antrl 741	. . . . 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 1166	. . . . . . 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 487	. . . . . 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 487	. . . . 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 8028	. . . . 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 596	. . . 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 1242	. . . 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 521	. . 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 4600	. . . . . 6 ⊢ ⟨𝑋, 𝑌, 𝑍⟩ = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩
11	10	eqeq2i 2778	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ 𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩)
12		eqop 8034	. . . . 5 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ↔ ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍)))
13	11, 12	bitrid 286	. . . 4 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍)))
14	13	ad2antrl 741	. . 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 260	. 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 5700	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
17	16	3adant3 1150	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
18		simp3 1156	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ 𝑊)
19	17, 18	opelxpd 5702	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
20	10, 19	eqeltrid 2869	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
21	20	adantr 486	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
22		eleq1 2853	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
23	22	adantl 487	. . . 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 260	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → 𝐴 ∈ ((𝑈 × 𝑉) × 𝑊))
25		2fveq3 6890	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (1^st ‘(1^st ‘𝐴)) = (1^st ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)))
26		ot1stg 8006	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (1^st ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑋)
27	25, 26	sylan9eqr 2822	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (1^st ‘(1^st ‘𝐴)) = 𝑋)
28		2fveq3 6890	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2^nd ‘(1^st ‘𝐴)) = (2^nd ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)))
29		ot2ndg 8007	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (2^nd ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑌)
30	28, 29	sylan9eqr 2822	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2^nd ‘(1^st ‘𝐴)) = 𝑌)
31		fveq2 6885	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2^nd ‘𝐴) = (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩))
32		ot3rdg 8008	. . . . . 6 ⊢ (𝑍 ∈ 𝑊 → (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
33	32	3ad2ant3 1153	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
34	31, 33	sylan9eqr 2822	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2^nd ‘𝐴) = 𝑍)
35	27, 30, 34	3jca 1146	. . 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 521	. 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 813	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ⟨cop 4597 ⟨cotp 4599 × cxp 5661 ‘cfv 6540 1^st c1st 7990 2^nd c2nd 7991

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pr 5406 ax-un 7742

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-sn 4592 df-pr 4594 df-op 4598 df-ot 4600 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-iota 6496 df-fun 6542 df-fv 6548 df-1st 7992 df-2nd 7993

This theorem is used by: (None)

W3C validator