MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isf34lem1 Structured version   Visualization version   GIF version

Theorem isf34lem1 10138
Description: Lemma for isfin3-4 10148. (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 4050 . . . 4 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
32cbvmptv 5186 . . 3 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
41, 3eqtri 2766 . 2 𝐹 = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
5 difeq2 4050 . 2 (𝑎 = 𝑋 → (𝐴𝑎) = (𝐴𝑋))
6 elpw2g 5266 . . 3 (𝐴𝑉 → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
76biimpar 478 . 2 ((𝐴𝑉𝑋𝐴) → 𝑋 ∈ 𝒫 𝐴)
8 difexg 5249 . . 3 (𝐴𝑉 → (𝐴𝑋) ∈ V)
98adantr 481 . 2 ((𝐴𝑉𝑋𝐴) → (𝐴𝑋) ∈ V)
104, 5, 7, 9fvmptd3 6890 1 ((𝐴𝑉𝑋𝐴) → (𝐹𝑋) = (𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  Vcvv 3429  cdif 3883  wss 3886  𝒫 cpw 4533  cmpt 5156  cfv 6426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-iota 6384  df-fun 6428  df-fv 6434
This theorem is referenced by:  compssiso  10140  isf34lem4  10143  isf34lem7  10145  isf34lem6  10146
  Copyright terms: Public domain W3C validator