Theorem bj-diagval 32267

Description: Value of the diagonal. (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
bj-diagval	⊢ (𝐴 ∈ 𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

Proof of Theorem bj-diagval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3185	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		incom 3767	. . 3 ⊢ ((𝐴 × 𝐴) ∩ I ) = ( I ∩ (𝐴 × 𝐴))
3		sqxpexg 6861	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
4		inex1g 4729	. . . 4 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ I ) ∈ V)
5	3, 4	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 × 𝐴) ∩ I ) ∈ V)
6	2, 5	syl5eqelr 2693	. 2 ⊢ (𝐴 ∈ 𝑉 → ( I ∩ (𝐴 × 𝐴)) ∈ V)
7		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
8	7	sqxpeqd 5065	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
9	8	ineq2d 3776	. . 3 ⊢ (𝑥 = 𝐴 → ( I ∩ (𝑥 × 𝑥)) = ( I ∩ (𝐴 × 𝐴)))
10		df-bj-diag 32266	. . 3 ⊢ Diag = (𝑥 ∈ V ↦ ( I ∩ (𝑥 × 𝑥)))
11	9, 10	fvmptg 6189	. 2 ⊢ ((𝐴 ∈ V ∧ ( I ∩ (𝐴 × 𝐴)) ∈ V) → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
12	1, 6, 11	syl2anc 691	1 ⊢ (𝐴 ∈ 𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   I cid 4948   × cxp 5036  ‘cfv 5804  Diagcdiag2 32265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-bj-diag 32266

This theorem is referenced by:  bj-eldiag  32268  bj-eldiag2  32269