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Theorem isf34lem1 9482
Description: Lemma for isfin3-4 9492. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
compss.a 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
Assertion
Ref Expression
isf34lem1 ((𝐴𝑉𝑋𝐴) → (𝐹𝑋) = (𝐴𝑋))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hints:   𝐹(𝑥)   𝑋(𝑥)

Proof of Theorem isf34lem1
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 elpw2g 5019 . . 3 (𝐴𝑉 → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
21biimpar 470 . 2 ((𝐴𝑉𝑋𝐴) → 𝑋 ∈ 𝒫 𝐴)
3 difexg 5003 . . 3 (𝐴𝑉 → (𝐴𝑋) ∈ V)
43adantr 473 . 2 ((𝐴𝑉𝑋𝐴) → (𝐴𝑋) ∈ V)
5 difeq2 3920 . . 3 (𝑎 = 𝑋 → (𝐴𝑎) = (𝐴𝑋))
6 compss.a . . . 4 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
7 difeq2 3920 . . . . 5 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
87cbvmptv 4943 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
96, 8eqtri 2821 . . 3 𝐹 = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
105, 9fvmptg 6505 . 2 ((𝑋 ∈ 𝒫 𝐴 ∧ (𝐴𝑋) ∈ V) → (𝐹𝑋) = (𝐴𝑋))
112, 4, 10syl2anc 580 1 ((𝐴𝑉𝑋𝐴) → (𝐹𝑋) = (𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  cdif 3766  wss 3769  𝒫 cpw 4349  cmpt 4922  cfv 6101
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-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-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-iota 6064  df-fun 6103  df-fv 6109
This theorem is referenced by:  compssiso  9484  isf34lem4  9487  isf34lem7  9489  isf34lem6  9490
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