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Theorem pw2f1o2val 38283
Description: Function value of the pw2f1o2 38282 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 7310 . . 3 (𝑋 ∈ (2𝑜𝑚 𝐴) → 𝑋 ∈ V)
2 imaexg 7301 . . 3 (𝑋 ∈ V → (𝑋 “ {1𝑜}) ∈ V)
31, 2syl 17 . 2 (𝑋 ∈ (2𝑜𝑚 𝐴) → (𝑋 “ {1𝑜}) ∈ V)
4 cnveq 5464 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
54imaeq1d 5647 . . 3 (𝑥 = 𝑋 → (𝑥 “ {1𝑜}) = (𝑋 “ {1𝑜}))
6 pw2f1o2.f . . 3 𝐹 = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))
75, 6fvmptg 6469 . 2 ((𝑋 ∈ (2𝑜𝑚 𝐴) ∧ (𝑋 “ {1𝑜}) ∈ V) → (𝐹𝑋) = (𝑋 “ {1𝑜}))
83, 7mpdan 678 1 (𝑋 ∈ (2𝑜𝑚 𝐴) → (𝐹𝑋) = (𝑋 “ {1𝑜}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  Vcvv 3350  {csn 4334  cmpt 4888  ccnv 5276  cima 5280  cfv 6068  (class class class)co 6842  1𝑜c1o 7757  2𝑜c2o 7758  𝑚 cmap 8060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fv 6076
This theorem is referenced by:  pw2f1o2val2  38284
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