MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isf34lem2 Structured version   Visualization version   GIF version

Theorem isf34lem2 9789
Description: Lemma for isfin3-4 9798. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
compss.a 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
Assertion
Ref Expression
isf34lem2 (𝐴𝑉𝐹:𝒫 𝐴⟶𝒫 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem isf34lem2
StepHypRef Expression
1 difss 4112 . . . 4 (𝐴𝑥) ⊆ 𝐴
2 elpw2g 5244 . . . 4 (𝐴𝑉 → ((𝐴𝑥) ∈ 𝒫 𝐴 ↔ (𝐴𝑥) ⊆ 𝐴))
31, 2mpbiri 259 . . 3 (𝐴𝑉 → (𝐴𝑥) ∈ 𝒫 𝐴)
43adantr 481 . 2 ((𝐴𝑉𝑥 ∈ 𝒫 𝐴) → (𝐴𝑥) ∈ 𝒫 𝐴)
5 compss.a . 2 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
64, 5fmptd 6876 1 (𝐴𝑉𝐹:𝒫 𝐴⟶𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  cdif 3937  wss 3940  𝒫 cpw 4542  cmpt 5143  wf 6350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362
This theorem is referenced by:  isf34lem5  9794  isf34lem7  9795  isf34lem6  9796
  Copyright terms: Public domain W3C validator