Theorem bj-diagval 33090

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 3212	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		incom 3805	. . 3 ⊢ ((𝐴 × 𝐴) ∩ I ) = ( I ∩ (𝐴 × 𝐴))
3		sqxpexg 6963	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
4		inex1g 4801	. . . 4 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ I ) ∈ V)
5	3, 4	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 × 𝐴) ∩ I ) ∈ V)
6	2, 5	syl5eqelr 2706	. 2 ⊢ (𝐴 ∈ 𝑉 → ( I ∩ (𝐴 × 𝐴)) ∈ V)
7		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
8	7	sqxpeqd 5141	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
9	8	ineq2d 3814	. . 3 ⊢ (𝑥 = 𝐴 → ( I ∩ (𝑥 × 𝑥)) = ( I ∩ (𝐴 × 𝐴)))
10		df-bj-diag 33089	. . 3 ⊢ Diag = (𝑥 ∈ V ↦ ( I ∩ (𝑥 × 𝑥)))
11	9, 10	fvmptg 6280	. 2 ⊢ ((𝐴 ∈ V ∧ ( I ∩ (𝐴 × 𝐴)) ∈ V) → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
12	1, 6, 11	syl2anc 693	1 ⊢ (𝐴 ∈ 𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∩ cin 3573   I cid 5023   × cxp 5112  ‘cfv 5888  Diagcdiag2 33088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-bj-diag 33089

This theorem is referenced by:  bj-eldiag  33091  bj-eldiag2  33092