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Theorem pw2f1o2val 42460
Description: Function value of the pw2f1o2 42459 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypothesis
Ref Expression
pw2f1o2.f 𝐹 = (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o}))
Assertion
Ref Expression
pw2f1o2val (𝑋 ∈ (2om 𝐴) → (𝐹𝑋) = (𝑋 “ {1o}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem pw2f1o2val
StepHypRef Expression
1 cnvexg 7932 . . 3 (𝑋 ∈ (2om 𝐴) → 𝑋 ∈ V)
2 imaexg 7921 . . 3 (𝑋 ∈ V → (𝑋 “ {1o}) ∈ V)
31, 2syl 17 . 2 (𝑋 ∈ (2om 𝐴) → (𝑋 “ {1o}) ∈ V)
4 cnveq 5876 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
54imaeq1d 6062 . . 3 (𝑥 = 𝑋 → (𝑥 “ {1o}) = (𝑋 “ {1o}))
6 pw2f1o2.f . . 3 𝐹 = (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o}))
75, 6fvmptg 7003 . 2 ((𝑋 ∈ (2om 𝐴) ∧ (𝑋 “ {1o}) ∈ V) → (𝐹𝑋) = (𝑋 “ {1o}))
83, 7mpdan 686 1 (𝑋 ∈ (2om 𝐴) → (𝐹𝑋) = (𝑋 “ {1o}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3471  {csn 4629  cmpt 5231  ccnv 5677  cima 5681  cfv 6548  (class class class)co 7420  1oc1o 8480  2oc2o 8481  m cmap 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fv 6556
This theorem is referenced by:  pw2f1o2val2  42461
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