Theorem bj-imdirid 38375

Description: Functorial property of the direct image: the direct image by the identity on a set is the identity on the powerset. (Contributed by BJ, 24-Dec-2023.)

Hypothesis

Ref	Expression
bj-imdirid.ex	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Assertion

Ref	Expression
bj-imdirid	⊢ (𝜑 → ((𝐴 𝒫* 𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))

Proof of Theorem bj-imdirid

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-imdirid.ex	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		idssxp 6059	. . . 4 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
3	2	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
4	1, 1, 3	bj-imdirval2 38372	. 2 ⊢ (𝜑 → ((𝐴 𝒫* 𝐴)‘( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦)})
5		resiima 6092	. . . . 5 ⊢ (𝑥 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝑥) = 𝑥)
6	5	adantr 486	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (( I ↾ 𝐴) “ 𝑥) = 𝑥)
7	6	eqeq1d 2765	. . 3 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → ((( I ↾ 𝐴) “ 𝑥) = 𝑦 ↔ 𝑥 = 𝑦))
8	7	bj-imdiridlem 38374	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦)} = ( I ↾ 𝒫 𝐴)
9	4, 8	eqtrdi 2814	1 ⊢ (𝜑 → ((𝐴 𝒫* 𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  𝒫 cpw 4560  {copab 5173   I cid 5552   × cxp 5656   ↾ cres 5660   “ cima 5661  ‘cfv 6566  (class class class)co 7451   𝒫_* cimdir 38367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-imdir 38368

This theorem is used by: (None)