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Theorem bj-iminvid 34890
 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 (𝜑𝐴𝑈)
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.
StepHypRef Expression
1 bj-iminvid.ex . . 3 (𝜑𝐴𝑈)
2 idssxp 5888 . . . 4 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
32a1i 11 . . 3 (𝜑 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
41, 1, 3bj-iminvval2 34889 . 2 (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ 𝑥 = (( I ↾ 𝐴) “ 𝑦))})
5 cnvresid 6414 . . . . . . 7 ( I ↾ 𝐴) = ( I ↾ 𝐴)
65imaeq1i 5898 . . . . . 6 (( I ↾ 𝐴) “ 𝑦) = (( I ↾ 𝐴) “ 𝑦)
7 resiima 5916 . . . . . 6 (𝑦𝐴 → (( I ↾ 𝐴) “ 𝑦) = 𝑦)
86, 7syl5eq 2805 . . . . 5 (𝑦𝐴 → (( I ↾ 𝐴) “ 𝑦) = 𝑦)
98adantl 485 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴) “ 𝑦) = 𝑦)
109eqeq2d 2769 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑥 = (( I ↾ 𝐴) “ 𝑦) ↔ 𝑥 = 𝑦))
1110bj-imdiridlem 34880 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ 𝑥 = (( I ↾ 𝐴) “ 𝑦))} = ( I ↾ 𝒫 𝐴)
124, 11eqtrdi 2809 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 3858  𝒫 cpw 4494  {copab 5094   I cid 5429   × cxp 5522  ◡ccnv 5523   ↾ cres 5526   “ cima 5527  ‘cfv 6335  (class class class)co 7150  𝒫*ciminv 34886 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 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-iminv 34887 This theorem is referenced by: (None)
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