Theorem bj-iminvid 37900

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 6053	. . . 4 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
3	2	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
4	1, 1, 3	bj-iminvval2 37899	. 2 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = (◡( I ↾ 𝐴) “ 𝑦))})
5		cnvresid 6619	. . . . . . 7 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
6	5	imaeq1i 6061	. . . . . 6 ⊢ (◡( I ↾ 𝐴) “ 𝑦) = (( I ↾ 𝐴) “ 𝑦)
7		resiima 6080	. . . . . 6 ⊢ (𝑦 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝑦) = 𝑦)
8	6, 7	eqtrid 2812	. . . . 5 ⊢ (𝑦 ⊆ 𝐴 → (◡( I ↾ 𝐴) “ 𝑦) = 𝑦)
9	8	adantl 487	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (◡( I ↾ 𝐴) “ 𝑦) = 𝑦)
10	9	eqeq2d 2776	. . 3 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (𝑥 = (◡( I ↾ 𝐴) “ 𝑦) ↔ 𝑥 = 𝑦))
11	10	bj-imdiridlem 37890	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = (◡( I ↾ 𝐴) “ 𝑦))} = ( I ↾ 𝒫 𝐴)
12	4, 11	eqtrdi 2816	1 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906  𝒫 cpw 4564  {copab 5175   I cid 5557   × cxp 5661  ^◡ccnv 5662   ↾ cres 5665   “ cima 5666  ‘cfv 6540  (class class class)co 7419  𝒫^*ciminv 37896

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  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 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-iminv 37897

This theorem is used by: (None)