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Theorem bj-diagval 33322
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 3316 . 2 (𝐴𝑉𝐴 ∈ V)
2 incom 3913 . . 3 ((𝐴 × 𝐴) ∩ I ) = ( I ∩ (𝐴 × 𝐴))
3 sqxpexg 7080 . . . 4 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
4 inex1g 4909 . . . 4 ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ I ) ∈ V)
53, 4syl 17 . . 3 (𝐴𝑉 → ((𝐴 × 𝐴) ∩ I ) ∈ V)
62, 5syl5eqelr 2808 . 2 (𝐴𝑉 → ( I ∩ (𝐴 × 𝐴)) ∈ V)
7 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
87sqxpeqd 5250 . . . 4 (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
98ineq2d 3922 . . 3 (𝑥 = 𝐴 → ( I ∩ (𝑥 × 𝑥)) = ( I ∩ (𝐴 × 𝐴)))
10 df-bj-diag 33321 . . 3 Diag = (𝑥 ∈ V ↦ ( I ∩ (𝑥 × 𝑥)))
119, 10fvmptg 6394 . 2 ((𝐴 ∈ V ∧ ( I ∩ (𝐴 × 𝐴)) ∈ V) → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
121, 6, 11syl2anc 696 1 (𝐴𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  wcel 2103  Vcvv 3304  cin 3679   I cid 5127   × cxp 5216  cfv 6001  Diagcdiag2 33320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-iota 5964  df-fun 6003  df-fv 6009  df-bj-diag 33321
This theorem is referenced by:  bj-eldiag  33323  bj-eldiag2  33324
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