Theorem bj-iminvid 37859

Description: Functorial property of the inverse image: the inverse image by the identity on a set is the identity on the powerset. (Contributed by BJ, 26-May-2024.)

Hypothesis

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

Assertion

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

Proof of Theorem bj-iminvid

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

Step	Hyp	Ref	Expression
1		bj-iminvid.ex	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		idssxp 6051	. . . 4 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
3	2	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
4	1, 1, 3	bj-iminvval2 37858	. 2 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = (◡( I ↾ 𝐴) “ 𝑦))})
5		cnvresid 6615	. . . . . . 7 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
6	5	imaeq1i 6059	. . . . . 6 ⊢ (◡( I ↾ 𝐴) “ 𝑦) = (( I ↾ 𝐴) “ 𝑦)
7		resiima 6078	. . . . . 6 ⊢ (𝑦 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝑦) = 𝑦)
8	6, 7	eqtrid 2810	. . . . 5 ⊢ (𝑦 ⊆ 𝐴 → (◡( I ↾ 𝐴) “ 𝑦) = 𝑦)
9	8	adantl 486	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (◡( I ↾ 𝐴) “ 𝑦) = 𝑦)
10	9	eqeq2d 2774	. . 3 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (𝑥 = (◡( I ↾ 𝐴) “ 𝑦) ↔ 𝑥 = 𝑦))
11	10	bj-imdiridlem 37849	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = (◡( I ↾ 𝐴) “ 𝑦))} = ( I ↾ 𝒫 𝐴)
12	4, 11	eqtrdi 2814	1 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905  𝒫 cpw 4562  {copab 5173   I cid 5555   × cxp 5659  ^◡ccnv 5660   ↾ cres 5663   “ cima 5664  ‘cfv 6536  (class class class)co 7410  𝒫^*ciminv 37855

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 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

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 3745  df-csb 3854  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-iun 4958  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-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-iminv 37856

This theorem is referenced by: (None)