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Theorem bj-elid6 36558
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 5681 . . . 4 ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
21elin2 4192 . . 3 (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × V)))
32biancomi 462 . 2 (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ))
4 bj-elid4 36556 . . 3 (𝐵 ∈ (𝐴 × V) → (𝐵 ∈ I ↔ (1st𝐵) = (2nd𝐵)))
54pm5.32i 574 . 2 ((𝐵 ∈ (𝐴 × V) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × V) ∧ (1st𝐵) = (2nd𝐵)))
6 1st2nd2 8013 . . . . 5 (𝐵 ∈ (𝐴 × V) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
76pm4.71ri 560 . . . 4 (𝐵 ∈ (𝐴 × V) ↔ (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)))
8 eleq1 2815 . . . . . . . 8 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (𝐴 × V) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × V)))
98adantl 481 . . . . . . 7 (((1st𝐵) = (2nd𝐵) ∧ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩) → (𝐵 ∈ (𝐴 × V) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × V)))
10 simpl 482 . . . . . . . . . . . 12 (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V) → (1st𝐵) ∈ 𝐴)
1110a1i 11 . . . . . . . . . . 11 ((1st𝐵) = (2nd𝐵) → (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V) → (1st𝐵) ∈ 𝐴))
12 eleq1 2815 . . . . . . . . . . . 12 ((1st𝐵) = (2nd𝐵) → ((1st𝐵) ∈ 𝐴 ↔ (2nd𝐵) ∈ 𝐴))
1310, 12imbitrid 243 . . . . . . . . . . 11 ((1st𝐵) = (2nd𝐵) → (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V) → (2nd𝐵) ∈ 𝐴))
1411, 13jcad 512 . . . . . . . . . 10 ((1st𝐵) = (2nd𝐵) → (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V) → ((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ 𝐴)))
15 elex 3487 . . . . . . . . . . 11 ((2nd𝐵) ∈ 𝐴 → (2nd𝐵) ∈ V)
1615anim2i 616 . . . . . . . . . 10 (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ 𝐴) → ((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V))
1714, 16impbid1 224 . . . . . . . . 9 ((1st𝐵) = (2nd𝐵) → (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V) ↔ ((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ 𝐴)))
1817adantr 480 . . . . . . . 8 (((1st𝐵) = (2nd𝐵) ∧ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩) → (((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V) ↔ ((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ 𝐴)))
19 opelxp 5705 . . . . . . . 8 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × V) ↔ ((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ V))
20 opelxp 5705 . . . . . . . 8 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × 𝐴) ↔ ((1st𝐵) ∈ 𝐴 ∧ (2nd𝐵) ∈ 𝐴))
2118, 19, 203bitr4g 314 . . . . . . 7 (((1st𝐵) = (2nd𝐵) ∧ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩) → (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × V) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × 𝐴)))
22 eleq1 2815 . . . . . . . . 9 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (𝐴 × 𝐴) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × 𝐴)))
2322bicomd 222 . . . . . . . 8 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × 𝐴) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
2423adantl 481 . . . . . . 7 (((1st𝐵) = (2nd𝐵) ∧ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩) → (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (𝐴 × 𝐴) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
259, 21, 243bitrd 305 . . . . . 6 (((1st𝐵) = (2nd𝐵) ∧ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩) → (𝐵 ∈ (𝐴 × V) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
2625pm5.32da 578 . . . . 5 ((1st𝐵) = (2nd𝐵) → ((𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)) ↔ (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴))))
27 simpr 484 . . . . . 6 ((𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)) → 𝐵 ∈ (𝐴 × 𝐴))
28 1st2nd2 8013 . . . . . . 7 (𝐵 ∈ (𝐴 × 𝐴) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
2928ancri 549 . . . . . 6 (𝐵 ∈ (𝐴 × 𝐴) → (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)))
3027, 29impbii 208 . . . . 5 ((𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × 𝐴)) ↔ 𝐵 ∈ (𝐴 × 𝐴))
3126, 30bitrdi 287 . . . 4 ((1st𝐵) = (2nd𝐵) → ((𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ ∧ 𝐵 ∈ (𝐴 × V)) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
327, 31bitrid 283 . . 3 ((1st𝐵) = (2nd𝐵) → (𝐵 ∈ (𝐴 × V) ↔ 𝐵 ∈ (𝐴 × 𝐴)))
3332pm5.32ri 575 . 2 ((𝐵 ∈ (𝐴 × V) ∧ (1st𝐵) = (2nd𝐵)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵)))
343, 5, 333bitri 297 1 (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  cop 4629   I cid 5566   × cxp 5667  cres 5671  cfv 6537  1st c1st 7972  2nd c2nd 7973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6489  df-fun 6539  df-fv 6545  df-1st 7974  df-2nd 7975
This theorem is referenced by: (None)
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