bj-elid6 - Mathbox for BJ

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Theorem bj-elid6 34465

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 5567	. . . 4 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
2	1	elin2 4174	. . 3 ⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × V)))
3	2	biancomi 465	. 2 ⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ))
4		bj-elid4 34463	. . 3 ⊢ (𝐵 ∈ (𝐴 × V) → (𝐵 ∈ I ↔ (1^st ‘𝐵) = (2^nd ‘𝐵)))
5	4	pm5.32i 577	. 2 ⊢ ((𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × V) ∧ (1^st ‘𝐵) = (2^nd ‘𝐵)))
6		1st2nd2 7728	. . . . 5 ⊢ (𝐵 ∈ (𝐴 × V) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
7	6	pm4.71ri 563	. . . 4 ⊢ (𝐵 ∈ (𝐴 × V) ↔ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)))
8		eleq1 2900	. . . . . . . 8 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (𝐴 × V) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × V)))
9	8	adantl 484	. . . . . . 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 485	. . . . . . . . . . . 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 2900	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → ((1^st ‘𝐵) ∈ 𝐴 ↔ (2^nd ‘𝐵) ∈ 𝐴))
13	10, 12	syl5ib 246	. . . . . . . . . . 11 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → (((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V) → (2^nd ‘𝐵) ∈ 𝐴))
14	11, 13	jcad 515	. . . . . . . . . 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 3512	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝐵) ∈ 𝐴 → (2^nd ‘𝐵) ∈ V)
16	15	anim2i 618	. . . . . . . . . 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 483	. . . . . . . 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 5591	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (𝐴 × V) ↔ ((1^st ‘𝐵) ∈ 𝐴 ∧ (2^nd ‘𝐵) ∈ V))
20		opelxp 5591	. . . . . . . 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 2900	. . . . . . . . 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 484	. . . . . . 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 581	. . . . 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 487	. . . . . 6 ⊢ ((𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)) → 𝐵 ∈ (𝐴 × 𝐴))
28		1st2nd2 7728	. . . . . . 7 ⊢ (𝐵 ∈ (𝐴 × 𝐴) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
29	28	ancri 552	. . . . . 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	syl6bb 289	. . . 4 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → ((𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
32	7, 31	syl5bb 285	. . 3 ⊢ ((1^st ‘𝐵) = (2^nd ‘𝐵) → (𝐵 ∈ (𝐴 × V) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
33	32	pm5.32ri 578	. 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 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⟨cop 4573 I cid 5459 × cxp 5553 ↾ cres 5557 ‘cfv 6355 1^st c1st 7687 2^nd c2nd 7688

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-iota 6314 df-fun 6357 df-fv 6363 df-1st 7689 df-2nd 7690

This theorem is referenced by: (None)

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