Theorem bj-imdirid 37811

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 → ((𝐴𝒫*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 6053	. . . 4 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
3	2	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
4	1, 1, 3	bj-imdirval2 37808	. 2 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦)})
5		resiima 6080	. . . . 5 ⊢ (𝑥 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝑥) = 𝑥)
6	5	adantr 485	. . . 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 37810	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦)} = ( I ↾ 𝒫 𝐴)
9	4, 8	eqtrdi 2814	1 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ⊆ wss 3906  𝒫 cpw 4563  {copab 5174   I cid 5557   × cxp 5661   ↾ cres 5665   “ cima 5666  ‘cfv 6538  (class class class)co 7412  𝒫_*cimdir 37803

This theorem was proved from 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-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem 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-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-imdir 37804

This theorem is referenced by: (None)