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Theorem pw2f1o2val 42524
Description: Function value of the pw2f1o2 42523 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 7928 . . 3 (𝑋 ∈ (2om 𝐴) → 𝑋 ∈ V)
2 imaexg 7917 . . 3 (𝑋 ∈ V → (𝑋 “ {1o}) ∈ V)
31, 2syl 17 . 2 (𝑋 ∈ (2om 𝐴) → (𝑋 “ {1o}) ∈ V)
4 cnveq 5870 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
54imaeq1d 6057 . . 3 (𝑥 = 𝑋 → (𝑥 “ {1o}) = (𝑋 “ {1o}))
6 pw2f1o2.f . . 3 𝐹 = (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o}))
75, 6fvmptg 6997 . 2 ((𝑋 ∈ (2om 𝐴) ∧ (𝑋 “ {1o}) ∈ V) → (𝐹𝑋) = (𝑋 “ {1o}))
83, 7mpdan 685 1 (𝑋 ∈ (2om 𝐴) → (𝐹𝑋) = (𝑋 “ {1o}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3463  {csn 4624  cmpt 5226  ccnv 5671  cima 5675  cfv 6542  (class class class)co 7415  1oc1o 8476  2oc2o 8477  m cmap 8841
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 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
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 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550
This theorem is referenced by:  pw2f1o2val2  42525
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