Theorem bj-iminvid 37339

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 6006	. . . 4 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
3	2	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
4	1, 1, 3	bj-iminvval2 37338	. 2 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = (◡( I ↾ 𝐴) “ 𝑦))})
5		cnvresid 6569	. . . . . . 7 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
6	5	imaeq1i 6014	. . . . . 6 ⊢ (◡( I ↾ 𝐴) “ 𝑦) = (( I ↾ 𝐴) “ 𝑦)
7		resiima 6033	. . . . . 6 ⊢ (𝑦 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝑦) = 𝑦)
8	6, 7	eqtrid 2781	. . . . 5 ⊢ (𝑦 ⊆ 𝐴 → (◡( I ↾ 𝐴) “ 𝑦) = 𝑦)
9	8	adantl 481	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (◡( I ↾ 𝐴) “ 𝑦) = 𝑦)
10	9	eqeq2d 2745	. . 3 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (𝑥 = (◡( I ↾ 𝐴) “ 𝑦) ↔ 𝑥 = 𝑦))
11	10	bj-imdiridlem 37329	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = (◡( I ↾ 𝐴) “ 𝑦))} = ( I ↾ 𝒫 𝐴)
12	4, 11	eqtrdi 2785	1 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ⊆ wss 3899  𝒫 cpw 4552  {copab 5158   I cid 5516   × cxp 5620  ^◡ccnv 5621   ↾ cres 5624   “ cima 5625  ‘cfv 6490  (class class class)co 7356  𝒫^*ciminv 37335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-iminv 37336

This theorem is referenced by: (None)