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Theorem isf34lem1 9232
Description: Lemma for isfin3-4 9242. (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 4857 . . 3 (𝐴𝑉 → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
21biimpar 501 . 2 ((𝐴𝑉𝑋𝐴) → 𝑋 ∈ 𝒫 𝐴)
3 difexg 4841 . . 3 (𝐴𝑉 → (𝐴𝑋) ∈ V)
43adantr 480 . 2 ((𝐴𝑉𝑋𝐴) → (𝐴𝑋) ∈ V)
5 difeq2 3755 . . 3 (𝑎 = 𝑋 → (𝐴𝑎) = (𝐴𝑋))
6 compss.a . . . 4 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
7 difeq2 3755 . . . . 5 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
87cbvmptv 4783 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
96, 8eqtri 2673 . . 3 𝐹 = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
105, 9fvmptg 6319 . 2 ((𝑋 ∈ 𝒫 𝐴 ∧ (𝐴𝑋) ∈ V) → (𝐹𝑋) = (𝐴𝑋))
112, 4, 10syl2anc 694 1 ((𝐴𝑉𝑋𝐴) → (𝐹𝑋) = (𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  wss 3607  𝒫 cpw 4191  cmpt 4762  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  compssiso  9234  isf34lem4  9237  isf34lem7  9239  isf34lem6  9240
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