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Theorem pw2f1o2val 39626
Description: Function value of the pw2f1o2 39625 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 7621 . . 3 (𝑋 ∈ (2om 𝐴) → 𝑋 ∈ V)
2 imaexg 7612 . . 3 (𝑋 ∈ V → (𝑋 “ {1o}) ∈ V)
31, 2syl 17 . 2 (𝑋 ∈ (2om 𝐴) → (𝑋 “ {1o}) ∈ V)
4 cnveq 5737 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
54imaeq1d 5921 . . 3 (𝑥 = 𝑋 → (𝑥 “ {1o}) = (𝑋 “ {1o}))
6 pw2f1o2.f . . 3 𝐹 = (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o}))
75, 6fvmptg 6759 . 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 1531  wcel 2108  Vcvv 3493  {csn 4559  cmpt 5137  ccnv 5547  cima 5551  cfv 6348  (class class class)co 7148  1oc1o 8087  2oc2o 8088  m cmap 8398
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  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-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by:  pw2f1o2val2  39627
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