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Theorem isf34lem3 9080
Description: Lemma for isfin3-4 9087. (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 9076 . . 3 𝐹 = 𝐹
32imaeq1i 5382 . 2 (𝐹 “ (𝐹𝑋)) = (𝐹 “ (𝐹𝑋))
41compssiso 9079 . . . 4 (𝐴𝑉𝐹 Isom [] , [] (𝒫 𝐴, 𝒫 𝐴))
5 isof1o 6473 . . . 4 (𝐹 Isom [] , [] (𝒫 𝐴, 𝒫 𝐴) → 𝐹:𝒫 𝐴1-1-onto→𝒫 𝐴)
6 f1of1 6049 . . . 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 6066 . . 3 ((𝐹:𝒫 𝐴1-1→𝒫 𝐴𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹𝑋)) = 𝑋)
97, 8sylan 487 . 2 ((𝐴𝑉𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹𝑋)) = 𝑋)
103, 9syl5eqr 2658 1 ((𝐴𝑉𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  cdif 3537  wss 3540  𝒫 cpw 4108  cmpt 4643  ccnv 5037  cima 5041  1-1wf1 5801  1-1-ontowf1o 5803   Isom wiso 5805   [] crpss 6834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-rpss 6835
This theorem is referenced by:  isf34lem5  9083  isf34lem7  9084  isf34lem6  9085
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