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Theorem isf34lem1 9782
Description: Lemma for isfin3-4 9792. (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 4090 . . . 4 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
32cbvmptv 5160 . . 3 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
41, 3eqtri 2841 . 2 𝐹 = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
5 difeq2 4090 . 2 (𝑎 = 𝑋 → (𝐴𝑎) = (𝐴𝑋))
6 elpw2g 5238 . . 3 (𝐴𝑉 → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
76biimpar 478 . 2 ((𝐴𝑉𝑋𝐴) → 𝑋 ∈ 𝒫 𝐴)
8 difexg 5222 . . 3 (𝐴𝑉 → (𝐴𝑋) ∈ V)
98adantr 481 . 2 ((𝐴𝑉𝑋𝐴) → (𝐴𝑋) ∈ V)
104, 5, 7, 9fvmptd3 6783 1 ((𝐴𝑉𝑋𝐴) → (𝐹𝑋) = (𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cdif 3930  wss 3933  𝒫 cpw 4535  cmpt 5137  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by:  compssiso  9784  isf34lem4  9787  isf34lem7  9789  isf34lem6  9790
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