Theorem bj-iminvid 37196

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 6067	. . . 4 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
3	2	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
4	1, 1, 3	bj-iminvval2 37195	. 2 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = (◡( I ↾ 𝐴) “ 𝑦))})
5		cnvresid 6645	. . . . . . 7 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
6	5	imaeq1i 6075	. . . . . 6 ⊢ (◡( I ↾ 𝐴) “ 𝑦) = (( I ↾ 𝐴) “ 𝑦)
7		resiima 6094	. . . . . 6 ⊢ (𝑦 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝑦) = 𝑦)
8	6, 7	eqtrid 2789	. . . . 5 ⊢ (𝑦 ⊆ 𝐴 → (◡( I ↾ 𝐴) “ 𝑦) = 𝑦)
9	8	adantl 481	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (◡( I ↾ 𝐴) “ 𝑦) = 𝑦)
10	9	eqeq2d 2748	. . 3 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (𝑥 = (◡( I ↾ 𝐴) “ 𝑦) ↔ 𝑥 = 𝑦))
11	10	bj-imdiridlem 37186	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = (◡( I ↾ 𝐴) “ 𝑦))} = ( I ↾ 𝒫 𝐴)
12	4, 11	eqtrdi 2793	1 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  𝒫 cpw 4600  {copab 5205   I cid 5577   × cxp 5683  ^◡ccnv 5684   ↾ cres 5687   “ cima 5688  ‘cfv 6561  (class class class)co 7431  𝒫^*ciminv 37192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-iminv 37193

This theorem is referenced by: (None)