Theorem bj-diagval2 37847

Description: Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the diagonal function, whereas bj-diagval 37846 views it as the functionalized identity. See df-bj-diag 37845 for the terminology. (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
bj-diagval2	⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

Proof of Theorem bj-diagval2

Step	Hyp	Ref	Expression
1		bj-diagval 37846	. 2 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))
2		idinxpresid 6050	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
3	1, 2	eqtr4di 2816	1 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2143   ∩ cin 3904   I cid 5555   × cxp 5659   ↾ cres 5663  ‘cfv 6536  Idcdiag2 37844

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-bj-diag 37845

This theorem is used by:  bj-eldiag  37848  bj-eldiag2  37849