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Theorem bj-imdirid 36574
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 6042 . . . 4 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
32a1i 11 . . 3 (𝜑 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
41, 1, 3bj-imdirval2 36571 . 2 (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦)})
5 resiima 6069 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴) “ 𝑥) = 𝑥)
65adantr 480 . . . 4 ((𝑥𝐴𝑦𝐴) → (( I ↾ 𝐴) “ 𝑥) = 𝑥)
76eqeq1d 2728 . . 3 ((𝑥𝐴𝑦𝐴) → ((( I ↾ 𝐴) “ 𝑥) = 𝑦𝑥 = 𝑦))
87bj-imdiridlem 36573 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦)} = ( I ↾ 𝒫 𝐴)
94, 8eqtrdi 2782 1 (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wss 3943  𝒫 cpw 4597  {copab 5203   I cid 5566   × cxp 5667  cres 5671  cima 5672  cfv 6537  (class class class)co 7405  𝒫*cimdir 36566
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-imdir 36567
This theorem is referenced by: (None)
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