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Theorem pw2f1o2val 38449
 Description: Function value of the pw2f1o2 38448 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 7374 . . 3 (𝑋 ∈ (2o𝑚 𝐴) → 𝑋 ∈ V)
2 imaexg 7365 . . 3 (𝑋 ∈ V → (𝑋 “ {1o}) ∈ V)
31, 2syl 17 . 2 (𝑋 ∈ (2o𝑚 𝐴) → (𝑋 “ {1o}) ∈ V)
4 cnveq 5528 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
54imaeq1d 5706 . . 3 (𝑥 = 𝑋 → (𝑥 “ {1o}) = (𝑋 “ {1o}))
6 pw2f1o2.f . . 3 𝐹 = (𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o}))
75, 6fvmptg 6527 . 2 ((𝑋 ∈ (2o𝑚 𝐴) ∧ (𝑋 “ {1o}) ∈ V) → (𝐹𝑋) = (𝑋 “ {1o}))
83, 7mpdan 680 1 (𝑋 ∈ (2o𝑚 𝐴) → (𝐹𝑋) = (𝑋 “ {1o}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  Vcvv 3414  {csn 4397   ↦ cmpt 4952  ◡ccnv 5341   “ cima 5345  ‘cfv 6123  (class class class)co 6905  1oc1o 7819  2oc2o 7820   ↑𝑚 cmap 8122 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fv 6131 This theorem is referenced by:  pw2f1o2val2  38450
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