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Theorem bj-imdirid 34752
 Description: Functorial property of the direct image: the direct image by the identity on a set is the identity on the powerset. (Contributed by BJ, 24-Dec-2023.)
Hypothesis
Ref Expression
bj-imdirid.ex (𝜑𝐴𝑈)
Assertion
Ref Expression
bj-imdirid (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))

Proof of Theorem bj-imdirid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-imdirid.ex . . 3 (𝜑𝐴𝑈)
2 idssxp 5887 . . . 4 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
32a1i 11 . . 3 (𝜑 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
41, 1, 3bj-imdirval2 34749 . 2 (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦)})
5 resiima 5915 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴) “ 𝑥) = 𝑥)
65adantr 484 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴) “ 𝑥) = 𝑥)
76eqeq1d 2800 . . 3 ((𝑥𝐴𝑦𝐴) → ((( I ↾ 𝐴) “ 𝑥) = 𝑦𝑥 = 𝑦))
87bj-imdiridlem 34751 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦)} = ( I ↾ 𝒫 𝐴)
94, 8eqtrdi 2849 1 (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3883  𝒫 cpw 4500  {copab 5096   I cid 5428   × cxp 5521   ↾ cres 5525   “ cima 5526  ‘cfv 6332  (class class class)co 7145  𝒫*cimdir 34744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-imdir 34745 This theorem is referenced by: (None)
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