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Theorem isf34lem1 10402
Description: Lemma for isfin3-4 10412. (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 4112 . . . 4 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
32cbvmptv 5262 . . 3 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
41, 3eqtri 2753 . 2 𝐹 = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
5 difeq2 4112 . 2 (𝑎 = 𝑋 → (𝐴𝑎) = (𝐴𝑋))
6 elpw2g 5347 . . 3 (𝐴𝑉 → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
76biimpar 476 . 2 ((𝐴𝑉𝑋𝐴) → 𝑋 ∈ 𝒫 𝐴)
8 difexg 5330 . . 3 (𝐴𝑉 → (𝐴𝑋) ∈ V)
98adantr 479 . 2 ((𝐴𝑉𝑋𝐴) → (𝐴𝑋) ∈ V)
104, 5, 7, 9fvmptd3 7027 1 ((𝐴𝑉𝑋𝐴) → (𝐹𝑋) = (𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  cdif 3941  wss 3944  𝒫 cpw 4604  cmpt 5232  cfv 6549
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557
This theorem is referenced by:  compssiso  10404  isf34lem4  10407  isf34lem7  10409  isf34lem6  10410
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