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Theorem isf34lem1 9788
 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
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 4092 . . . 4 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
32cbvmptv 5161 . . 3 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
41, 3eqtri 2844 . 2 𝐹 = (𝑎 ∈ 𝒫 𝐴 ↦ (𝐴𝑎))
5 difeq2 4092 . 2 (𝑎 = 𝑋 → (𝐴𝑎) = (𝐴𝑋))
6 elpw2g 5239 . . 3 (𝐴𝑉 → (𝑋 ∈ 𝒫 𝐴𝑋𝐴))
76biimpar 480 . 2 ((𝐴𝑉𝑋𝐴) → 𝑋 ∈ 𝒫 𝐴)
8 difexg 5223 . . 3 (𝐴𝑉 → (𝐴𝑋) ∈ V)
98adantr 483 . 2 ((𝐴𝑉𝑋𝐴) → (𝐴𝑋) ∈ V)
104, 5, 7, 9fvmptd3 6785 1 ((𝐴𝑉𝑋𝐴) → (𝐹𝑋) = (𝐴𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ∖ cdif 3932   ⊆ wss 3935  𝒫 cpw 4538   ↦ cmpt 5138  ‘cfv 6349 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357 This theorem is referenced by:  compssiso  9790  isf34lem4  9793  isf34lem7  9795  isf34lem6  9796
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