bj-elid6 - Mathbox for BJ

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Theorem bj-elid6 37842

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 5672	. . . 4 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
2	1	elin2 4155	. . 3 ⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × V)))
3	2	biancomi 467	. 2 ⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ))
4		bj-elid4 37840	. . 3 ⊢ (𝐵 ∈ (𝐴 × V) → (𝐵 ∈ I ↔ (1^st ‘𝐵) = (2^nd ‘𝐵)))
5	4	pm5.32i 584	. 2 ⊢ ((𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × V) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)))
6		1st2nd2 8023	. . . . 5 ⊢ (𝐵 ∈ (𝐴 × V) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
7	6	pm4.71ri 569	. . . 4 ⊢ (𝐵 ∈ (𝐴 × V) ↔ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)))
8		eleq1 2850	. . . . . . . 8 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (𝐴 × V) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × V)))
9	8	adantl 486	. . . . . . 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 487	. . . . . . . . . . . 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 2850	. . . . . . . . . . . 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 521	. . . . . . . . . 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 3475	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝐵) ∈ 𝐴 → (2^nd ‘𝐵) ∈ V)
16	15	anim2i 628	. . . . . . . . . 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 485	. . . . . . . 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 5696	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × V) ↔ ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V))
20		opelxp 5696	. . . . . . . 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 2850	. . . . . . . . 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 486	. . . . . . 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 589	. . . . 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 489	. . . . . 6 ⊢ ((𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)) → 𝐵 ∈ (𝐴 × 𝐴))
28		1st2nd2 8023	. . . . . . 7 ⊢ (𝐵 ∈ (𝐴 × 𝐴) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
29	28	ancri 558	. . . . . 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 585	. 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 400 = wceq 1569 ∈ wcel 2142 Vcvv 3454 ⟨cop 4594 I cid 5554 × cxp 5658 ↾ cres 5662 ‘cfv 6536 1^st c1st 7982 2^nd c2nd 7983

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pr 5403 ax-un 7734

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3416 df-v 3456 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5555 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-iota 6492 df-fun 6538 df-fv 6544 df-1st 7984 df-2nd 7985

This theorem is used by: (None)

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