Theorem bj-iminvid 34875

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 5881	. . . 4 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
3	2	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
4	1, 1, 3	bj-iminvval2 34874	. 2 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = (◡( I ↾ 𝐴) “ 𝑦))})
5		cnvresid 6407	. . . . . . 7 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
6	5	imaeq1i 5891	. . . . . 6 ⊢ (◡( I ↾ 𝐴) “ 𝑦) = (( I ↾ 𝐴) “ 𝑦)
7		resiima 5909	. . . . . 6 ⊢ (𝑦 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝑦) = 𝑦)
8	6, 7	syl5eq 2806	. . . . 5 ⊢ (𝑦 ⊆ 𝐴 → (◡( I ↾ 𝐴) “ 𝑦) = 𝑦)
9	8	adantl 486	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (◡( I ↾ 𝐴) “ 𝑦) = 𝑦)
10	9	eqeq2d 2770	. . 3 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (𝑥 = (◡( I ↾ 𝐴) “ 𝑦) ↔ 𝑥 = 𝑦))
11	10	bj-imdiridlem 34865	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = (◡( I ↾ 𝐴) “ 𝑦))} = ( I ↾ 𝒫 𝐴)
12	4, 11	eqtrdi 2810	1 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ⊆ wss 3854  𝒫 cpw 4487  {copab 5087   I cid 5422   × cxp 5515  ^◡ccnv 5516   ↾ cres 5519   “ cima 5520  ‘cfv 6328  (class class class)co 7143  𝒫^*ciminv 34871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5149  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-reu 3075  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-op 4522  df-uni 4792  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-id 5423  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7146  df-oprab 7147  df-mpo 7148  df-iminv 34872

This theorem is referenced by: (None)