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Theorem isf34lem3 9485
Description: Lemma for isfin3-4 9492. (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 9481 . . 3 𝐹 = 𝐹
32imaeq1i 5680 . 2 (𝐹 “ (𝐹𝑋)) = (𝐹 “ (𝐹𝑋))
41compssiso 9484 . . . 4 (𝐴𝑉𝐹 Isom [] , [] (𝒫 𝐴, 𝒫 𝐴))
5 isof1o 6801 . . . 4 (𝐹 Isom [] , [] (𝒫 𝐴, 𝒫 𝐴) → 𝐹:𝒫 𝐴1-1-onto→𝒫 𝐴)
6 f1of1 6355 . . . 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 6372 . . 3 ((𝐹:𝒫 𝐴1-1→𝒫 𝐴𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹𝑋)) = 𝑋)
97, 8sylan 576 . 2 ((𝐴𝑉𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹𝑋)) = 𝑋)
103, 9syl5eqr 2847 1 ((𝐴𝑉𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  cdif 3766  wss 3769  𝒫 cpw 4349  cmpt 4922  ccnv 5311  cima 5315  1-1wf1 6098  1-1-ontowf1o 6100   Isom wiso 6102   [] crpss 7170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-rpss 7171
This theorem is referenced by:  isf34lem5  9488  isf34lem7  9489  isf34lem6  9490
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