bj-elid6 - Mathbox for BJ

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Theorem bj-elid6 37922

Description: Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)

**Assertion**
Ref	Expression
bj-elid6	⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)))

**Proof of Theorem bj-elid6**
Step	Hyp	Ref	Expression
1		df-res 5667	. . . 4 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
2	1	elin2 4149	. . 3 ⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × V)))
3	2	biancomi 468	. 2 ⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ))
4		bj-elid4 37920	. . 3 ⊢ (𝐵 ∈ (𝐴 × V) → (𝐵 ∈ I ↔ (1^st ‘𝐵) = (2^nd ‘𝐵)))
5	4	pm5.32i 585	. 2 ⊢ ((𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × V) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)))
6		1st2nd2 8025	. . . . 5 ⊢ (𝐵 ∈ (𝐴 × V) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
7	6	pm4.71ri 570	. . . 4 ⊢ (𝐵 ∈ (𝐴 × V) ↔ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)))
8		eleq1 2848	. . . . . . . 8 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (𝐴 × V) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × V)))
9	8	adantl 487	. . . . . . 7 ⊢ (((1^st ‘𝐵) = (2^nd ‘𝐵) ∧ 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩) → (𝐵 ∈ (𝐴 × V) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × V)))
10		simpl 488	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V) → (1^st ‘𝐵) ∈ 𝐴)
11	10	a1i 11	. . . . . . . . . . 11 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → (((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V) → (1^st ‘𝐵) ∈ 𝐴))
12		eleq1 2848	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → ((1^st ‘𝐵) ∈ 𝐴 ↔ (2^nd ‘𝐵) ∈ 𝐴))
13	10, 12	imbitrid 247	. . . . . . . . . . 11 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → (((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V) → (2^nd ‘𝐵) ∈ 𝐴))
14	11, 13	jcad 522	. . . . . . . . . 10 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → (((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V) → ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ 𝐴)))
15		elex 3471	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝐵) ∈ 𝐴 → (2^nd ‘𝐵) ∈ V)
16	15	anim2i 629	. . . . . . . . . 10 ⊢ (((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ 𝐴) → ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V))
17	14, 16	impbid1 228	. . . . . . . . 9 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → (((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V) ↔ ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ 𝐴)))
18	17	adantr 486	. . . . . . . 8 ⊢ (((1^st ‘𝐵) = (2^nd ‘𝐵) ∧ 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩) → (((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V) ↔ ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ 𝐴)))
19		opelxp 5691	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × V) ↔ ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V))
20		opelxp 5691	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × 𝐴) ↔ ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ 𝐴))
21	18, 19, 20	3bitr4g 317	. . . . . . 7 ⊢ (((1^st ‘𝐵) = (2^nd ‘𝐵) ∧ 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩) → (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × V) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × 𝐴)))
22		eleq1 2848	. . . . . . . . 9 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (𝐴 × 𝐴) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × 𝐴)))
23	22	bicomd 226	. . . . . . . 8 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × 𝐴) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
24	23	adantl 487	. . . . . . 7 ⊢ (((1^st ‘𝐵) = (2^nd ‘𝐵) ∧ 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩) → (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × 𝐴) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
25	9, 21, 24	3bitrd 308	. . . . . 6 ⊢ (((1^st ‘𝐵) = (2^nd ‘𝐵) ∧ 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩) → (𝐵 ∈ (𝐴 × V) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
26	25	pm5.32da 590	. . . . 5 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → ((𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)) ↔ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴))))
27		simpr 490	. . . . . 6 ⊢ ((𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)) → 𝐵 ∈ (𝐴 × 𝐴))
28		1st2nd2 8025	. . . . . . 7 ⊢ (𝐵 ∈ (𝐴 × 𝐴) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
29	28	ancri 559	. . . . . 6 ⊢ (𝐵 ∈ (𝐴 × 𝐴) → (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)))
30	27, 29	impbii 212	. . . . 5 ⊢ ((𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)) ↔ 𝐵 ∈ (𝐴 × 𝐴))
31	26, 30	bitrdi 290	. . . 4 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → ((𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
32	7, 31	bitrid 286	. . 3 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → (𝐵 ∈ (𝐴 × V) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
33	32	pm5.32ri 586	. 2 ⊢ ((𝐵 ∈ (𝐴 × V) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)))
34	3, 5, 33	3bitri 300	1 ⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3450 ⟨cop 4590 I cid 5549 × cxp 5653 ↾ cres 5657 ‘cfv 6533 1^st c1st 7984 2^nd c2nd 7985

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pr 5398 ax-un 7736

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-iota 6489 df-fun 6535 df-fv 6541 df-1st 7986 df-2nd 7987

This theorem is used by: (None)

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