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Theorem bj-diagval 33220
Description: Value of the diagonal. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-diagval (𝐴𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

Proof of Theorem bj-diagval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elex 3243 . 2 (𝐴𝑉𝐴 ∈ V)
2 incom 3838 . . 3 ((𝐴 × 𝐴) ∩ I ) = ( I ∩ (𝐴 × 𝐴))
3 sqxpexg 7005 . . . 4 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
4 inex1g 4834 . . . 4 ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ I ) ∈ V)
53, 4syl 17 . . 3 (𝐴𝑉 → ((𝐴 × 𝐴) ∩ I ) ∈ V)
62, 5syl5eqelr 2735 . 2 (𝐴𝑉 → ( I ∩ (𝐴 × 𝐴)) ∈ V)
7 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
87sqxpeqd 5175 . . . 4 (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
98ineq2d 3847 . . 3 (𝑥 = 𝐴 → ( I ∩ (𝑥 × 𝑥)) = ( I ∩ (𝐴 × 𝐴)))
10 df-bj-diag 33219 . . 3 Diag = (𝑥 ∈ V ↦ ( I ∩ (𝑥 × 𝑥)))
119, 10fvmptg 6319 . 2 ((𝐴 ∈ V ∧ ( I ∩ (𝐴 × 𝐴)) ∈ V) → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
121, 6, 11syl2anc 694 1 (𝐴𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606   I cid 5052   × cxp 5141  cfv 5926  Diagcdiag2 33218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-bj-diag 33219
This theorem is referenced by:  bj-eldiag  33221  bj-eldiag2  33222
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