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Theorem bj-elid6 37545

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 5633	. . . 4 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
2	1	elin2 4135	. . 3 ⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × V)))
3	2	biancomi 464	. 2 ⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ))
4		bj-elid4 37543	. . 3 ⊢ (𝐵 ∈ (𝐴 × V) → (𝐵 ∈ I ↔ (1^st ‘𝐵) = (2^nd ‘𝐵)))
5	4	pm5.32i 580	. 2 ⊢ ((𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × V) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)))
6		1st2nd2 7974	. . . . 5 ⊢ (𝐵 ∈ (𝐴 × V) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
7	6	pm4.71ri 566	. . . 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 483	. . . . . . 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 484	. . . . . . . . . . . 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 246	. . . . . . . . . . 11 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → (((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V) → (2^nd ‘𝐵) ∈ 𝐴))
14	11, 13	jcad 518	. . . . . . . . . 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 3454	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝐵) ∈ 𝐴 → (2^nd ‘𝐵) ∈ V)
16	15	anim2i 624	. . . . . . . . . 10 ⊢ (((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ 𝐴) → ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V))
17	14, 16	impbid1 227	. . . . . . . . 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 482	. . . . . . . 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 5657	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × V) ↔ ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V))
20		opelxp 5657	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × 𝐴) ↔ ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ 𝐴))
21	18, 19, 20	3bitr4g 316	. . . . . . 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 225	. . . . . . . 8 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × 𝐴) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
24	23	adantl 483	. . . . . . 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 307	. . . . . 6 ⊢ (((1^st ‘𝐵) = (2^nd ‘𝐵) ∧ 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩) → (𝐵 ∈ (𝐴 × V) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
26	25	pm5.32da 585	. . . . 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 486	. . . . . 6 ⊢ ((𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)) → 𝐵 ∈ (𝐴 × 𝐴))
28		1st2nd2 7974	. . . . . . 7 ⊢ (𝐵 ∈ (𝐴 × 𝐴) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
29	28	ancri 555	. . . . . 6 ⊢ (𝐵 ∈ (𝐴 × 𝐴) → (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)))
30	27, 29	impbii 211	. . . . 5 ⊢ ((𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)) ↔ 𝐵 ∈ (𝐴 × 𝐴))
31	26, 30	bitrdi 289	. . . 4 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → ((𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
32	7, 31	bitrid 285	. . 3 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → (𝐵 ∈ (𝐴 × V) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
33	32	pm5.32ri 581	. 2 ⊢ ((𝐵 ∈ (𝐴 × V) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)))
34	3, 5, 33	3bitri 299	1 ⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 397 = wceq 1548 ∈ wcel 2121 Vcvv 3433 ⟨cop 4564 I cid 5515 × cxp 5619 ↾ cres 5623 ‘cfv 6489 1^st c1st 7933 2^nd c2nd 7934

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5221 ax-nul 5231 ax-pr 5365 ax-un 7682

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-iota 6445 df-fun 6491 df-fv 6497 df-1st 7935 df-2nd 7936

This theorem is referenced by: (None)

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