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Theorem bj-imdirid 36721
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 6047 . . . 4 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
32a1i 11 . . 3 (𝜑 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
41, 1, 3bj-imdirval2 36718 . 2 (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦)})
5 resiima 6074 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴) “ 𝑥) = 𝑥)
65adantr 479 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴) “ 𝑥) = 𝑥)
76eqeq1d 2727 . . 3 ((𝑥𝐴𝑦𝐴) → ((( I ↾ 𝐴) “ 𝑥) = 𝑦𝑥 = 𝑦))
87bj-imdiridlem 36720 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦)} = ( I ↾ 𝒫 𝐴)
94, 8eqtrdi 2781 1 (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wss 3940  𝒫 cpw 4598  {copab 5205   I cid 5569   × cxp 5670  cres 5674  cima 5675  cfv 6542  (class class class)co 7415  𝒫*cimdir 36713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-imdir 36714
This theorem is referenced by: (None)
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