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Theorem bj-elid6 37171

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 5697	. . . 4 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
2	1	elin2 4203	. . 3 ⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × V)))
3	2	biancomi 462	. 2 ⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ))
4		bj-elid4 37169	. . 3 ⊢ (𝐵 ∈ (𝐴 × V) → (𝐵 ∈ I ↔ (1^st ‘𝐵) = (2^nd ‘𝐵)))
5	4	pm5.32i 574	. 2 ⊢ ((𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × V) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)))
6		1st2nd2 8053	. . . . 5 ⊢ (𝐵 ∈ (𝐴 × V) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
7	6	pm4.71ri 560	. . . 4 ⊢ (𝐵 ∈ (𝐴 × V) ↔ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)))
8		eleq1 2829	. . . . . . . 8 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (𝐴 × V) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × V)))
9	8	adantl 481	. . . . . . 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 482	. . . . . . . . . . . 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 2829	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → ((1^st ‘𝐵) ∈ 𝐴 ↔ (2^nd ‘𝐵) ∈ 𝐴))
13	10, 12	imbitrid 244	. . . . . . . . . . 11 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → (((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V) → (2^nd ‘𝐵) ∈ 𝐴))
14	11, 13	jcad 512	. . . . . . . . . 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 3501	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝐵) ∈ 𝐴 → (2^nd ‘𝐵) ∈ V)
16	15	anim2i 617	. . . . . . . . . 10 ⊢ (((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ 𝐴) → ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V))
17	14, 16	impbid1 225	. . . . . . . . 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 480	. . . . . . . 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 5721	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × V) ↔ ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V))
20		opelxp 5721	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × 𝐴) ↔ ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ 𝐴))
21	18, 19, 20	3bitr4g 314	. . . . . . 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 2829	. . . . . . . . 9 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (𝐴 × 𝐴) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × 𝐴)))
23	22	bicomd 223	. . . . . . . 8 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × 𝐴) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
24	23	adantl 481	. . . . . . 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 305	. . . . . 6 ⊢ (((1^st ‘𝐵) = (2^nd ‘𝐵) ∧ 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩) → (𝐵 ∈ (𝐴 × V) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
26	25	pm5.32da 579	. . . . 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 484	. . . . . 6 ⊢ ((𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)) → 𝐵 ∈ (𝐴 × 𝐴))
28		1st2nd2 8053	. . . . . . 7 ⊢ (𝐵 ∈ (𝐴 × 𝐴) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
29	28	ancri 549	. . . . . 6 ⊢ (𝐵 ∈ (𝐴 × 𝐴) → (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)))
30	27, 29	impbii 209	. . . . 5 ⊢ ((𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)) ↔ 𝐵 ∈ (𝐴 × 𝐴))
31	26, 30	bitrdi 287	. . . 4 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → ((𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
32	7, 31	bitrid 283	. . 3 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → (𝐵 ∈ (𝐴 × V) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
33	32	pm5.32ri 575	. 2 ⊢ ((𝐵 ∈ (𝐴 × V) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)))
34	3, 5, 33	3bitri 297	1 ⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⟨cop 4632 I cid 5577 × cxp 5683 ↾ cres 5687 ‘cfv 6561 1^st c1st 8012 2^nd c2nd 8013

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015

This theorem is referenced by: (None)

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