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Theorem isf34lem3 9994
Description: Lemma for isfin3-4 10001. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
compss.a 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
Assertion
Ref Expression
isf34lem3 ((𝐴𝑉𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹𝑋)) = 𝑋)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hints:   𝐹(𝑥)   𝑋(𝑥)

Proof of Theorem isf34lem3
StepHypRef Expression
1 compss.a . . . 4 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
21compsscnv 9990 . . 3 𝐹 = 𝐹
32imaeq1i 5931 . 2 (𝐹 “ (𝐹𝑋)) = (𝐹 “ (𝐹𝑋))
41compssiso 9993 . . . 4 (𝐴𝑉𝐹 Isom [] , [] (𝒫 𝐴, 𝒫 𝐴))
5 isof1o 7137 . . . 4 (𝐹 Isom [] , [] (𝒫 𝐴, 𝒫 𝐴) → 𝐹:𝒫 𝐴1-1-onto→𝒫 𝐴)
6 f1of1 6665 . . . 4 (𝐹:𝒫 𝐴1-1-onto→𝒫 𝐴𝐹:𝒫 𝐴1-1→𝒫 𝐴)
74, 5, 63syl 18 . . 3 (𝐴𝑉𝐹:𝒫 𝐴1-1→𝒫 𝐴)
8 f1imacnv 6682 . . 3 ((𝐹:𝒫 𝐴1-1→𝒫 𝐴𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹𝑋)) = 𝑋)
97, 8sylan 583 . 2 ((𝐴𝑉𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹𝑋)) = 𝑋)
103, 9eqtr3id 2792 1 ((𝐴𝑉𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  cdif 3868  wss 3871  𝒫 cpw 4518  cmpt 5140  ccnv 5555  cima 5559  1-1wf1 6382  1-1-ontowf1o 6384   Isom wiso 6386   [] crpss 7515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pr 5327
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-pss 3890  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-br 5059  df-opab 5121  df-mpt 5141  df-id 5460  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-res 5568  df-ima 5569  df-iota 6343  df-fun 6387  df-fn 6388  df-f 6389  df-f1 6390  df-fo 6391  df-f1o 6392  df-fv 6393  df-isom 6394  df-rpss 7516
This theorem is referenced by:  isf34lem5  9997  isf34lem7  9998  isf34lem6  9999
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