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Theorem bj-elid6 34454
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) ∧ (1st𝐵) = (2nd𝐵)))

Proof of Theorem bj-elid6
StepHypRef Expression
1 df-res 5560 . . . 4 ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
21elin2 4172 . . 3 (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × V)))
32biancomi 465 . 2 (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ))
4 bj-elid4 34452 . . 3 (𝐵 ∈ (𝐴 × V) → (𝐵 ∈ I ↔ (1st𝐵) = (2nd𝐵)))
54pm5.32i 577 . 2 ((𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × V) ∧ (1st𝐵) = (2nd𝐵)))
6 1st2nd2 7720 . . . . 5 (𝐵 ∈ (𝐴 × V) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
76pm4.71ri 563 . . . 4 (𝐵 ∈ (𝐴 × V) ↔ (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)))
8 eleq1 2898 . . . . . . . 8 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (𝐴 × V) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × V)))
98adantl 484 . . . . . . 7 (((1st𝐵) = (2nd𝐵) ∧ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩) → (𝐵 ∈ (𝐴 × V) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × V)))
10 simpl 485 . . . . . . . . . . . 12 (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V) → (1st𝐵) ∈ 𝐴)
1110a1i 11 . . . . . . . . . . 11 ((1st𝐵) = (2nd𝐵) → (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V) → (1st𝐵) ∈ 𝐴))
12 eleq1 2898 . . . . . . . . . . . 12 ((1st𝐵) = (2nd𝐵) → ((1st𝐵) ∈ 𝐴 ↔ (2nd𝐵) ∈ 𝐴))
1310, 12syl5ib 246 . . . . . . . . . . 11 ((1st𝐵) = (2nd𝐵) → (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V) → (2nd𝐵) ∈ 𝐴))
1411, 13jcad 515 . . . . . . . . . 10 ((1st𝐵) = (2nd𝐵) → (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V) → ((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ 𝐴)))
15 elex 3511 . . . . . . . . . . 11 ((2nd𝐵) ∈ 𝐴 → (2nd𝐵) ∈ V)
1615anim2i 618 . . . . . . . . . 10 (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ 𝐴) → ((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V))
1714, 16impbid1 227 . . . . . . . . 9 ((1st𝐵) = (2nd𝐵) → (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V) ↔ ((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ 𝐴)))
1817adantr 483 . . . . . . . 8 (((1st𝐵) = (2nd𝐵) ∧ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩) → (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V) ↔ ((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ 𝐴)))
19 opelxp 5584 . . . . . . . 8 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × V) ↔ ((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V))
20 opelxp 5584 . . . . . . . 8 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × 𝐴) ↔ ((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ 𝐴))
2118, 19, 203bitr4g 316 . . . . . . 7 (((1st𝐵) = (2nd𝐵) ∧ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩) → (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × V) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × 𝐴)))
22 eleq1 2898 . . . . . . . . 9 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (𝐴 × 𝐴) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × 𝐴)))
2322bicomd 225 . . . . . . . 8 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × 𝐴) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
2423adantl 484 . . . . . . 7 (((1st𝐵) = (2nd𝐵) ∧ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩) → (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × 𝐴) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
259, 21, 243bitrd 307 . . . . . 6 (((1st𝐵) = (2nd𝐵) ∧ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩) → (𝐵 ∈ (𝐴 × V) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
2625pm5.32da 581 . . . . 5 ((1st𝐵) = (2nd𝐵) → ((𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)) ↔ (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴))))
27 simpr 487 . . . . . 6 ((𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)) → 𝐵 ∈ (𝐴 × 𝐴))
28 1st2nd2 7720 . . . . . . 7 (𝐵 ∈ (𝐴 × 𝐴) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
2928ancri 552 . . . . . 6 (𝐵 ∈ (𝐴 × 𝐴) → (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)))
3027, 29impbii 211 . . . . 5 ((𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)) ↔ 𝐵 ∈ (𝐴 × 𝐴))
3126, 30syl6bb 289 . . . 4 ((1st𝐵) = (2nd𝐵) → ((𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
327, 31syl5bb 285 . . 3 ((1st𝐵) = (2nd𝐵) → (𝐵 ∈ (𝐴 × V) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
3332pm5.32ri 578 . 2 ((𝐵 ∈ (𝐴 × V) ∧ (1st𝐵) = (2nd𝐵)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵)))
343, 5, 333bitri 299 1 (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1531  wcel 2108  Vcvv 3493  cop 4565   I cid 5452   × cxp 5546  cres 5550  cfv 6348  1st c1st 7679  2nd c2nd 7680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-1st 7681  df-2nd 7682
This theorem is referenced by: (None)
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