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Theorem isf34lem1 10409
Description: Lemma for isfin3-4 10419. (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 compss.a . . 3 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
2 difeq2 4129 . . . 4 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
32cbvmptv 5260 . . 3 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
41, 3eqtri 2762 . 2 𝐹 = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
5 difeq2 4129 . 2 (𝑎 = 𝑋 → (𝐴𝑎) = (𝐴𝑋))
6 elpw2g 5338 . . 3 (𝐴𝑉 → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
76biimpar 477 . 2 ((𝐴𝑉𝑋𝐴) → 𝑋 ∈ 𝒫 𝐴)
8 difexg 5334 . . 3 (𝐴𝑉 → (𝐴𝑋) ∈ V)
98adantr 480 . 2 ((𝐴𝑉𝑋𝐴) → (𝐴𝑋) ∈ V)
104, 5, 7, 9fvmptd3 7038 1 ((𝐴𝑉𝑋𝐴) → (𝐹𝑋) = (𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  cdif 3959  wss 3962  𝒫 cpw 4604  cmpt 5230  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570
This theorem is referenced by:  compssiso  10411  isf34lem4  10414  isf34lem7  10416  isf34lem6  10417
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