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Theorem isf34lem1 9378
Description: Lemma for isfin3-4 9388. (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 4968 . . 3 (𝐴𝑉 → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
21biimpar 503 . 2 ((𝐴𝑉𝑋𝐴) → 𝑋 ∈ 𝒫 𝐴)
3 difexg 4952 . . 3 (𝐴𝑉 → (𝐴𝑋) ∈ V)
43adantr 472 . 2 ((𝐴𝑉𝑋𝐴) → (𝐴𝑋) ∈ V)
5 difeq2 3857 . . 3 (𝑎 = 𝑋 → (𝐴𝑎) = (𝐴𝑋))
6 compss.a . . . 4 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
7 difeq2 3857 . . . . 5 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
87cbvmptv 4894 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
96, 8eqtri 2774 . . 3 𝐹 = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
105, 9fvmptg 6434 . 2 ((𝑋 ∈ 𝒫 𝐴 ∧ (𝐴𝑋) ∈ V) → (𝐹𝑋) = (𝐴𝑋))
112, 4, 10syl2anc 696 1 ((𝐴𝑉𝑋𝐴) → (𝐹𝑋) = (𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  Vcvv 3332  cdif 3704  wss 3707  𝒫 cpw 4294  cmpt 4873  cfv 6041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049
This theorem is referenced by:  compssiso  9380  isf34lem4  9383  isf34lem7  9385  isf34lem6  9386
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