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Theorem pw2f1o2val 37125
Description: Function value of the pw2f1o2 37124 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 7074 . . 3 (𝑋 ∈ (2𝑜𝑚 𝐴) → 𝑋 ∈ V)
2 imaexg 7065 . . 3 (𝑋 ∈ V → (𝑋 “ {1𝑜}) ∈ V)
31, 2syl 17 . 2 (𝑋 ∈ (2𝑜𝑚 𝐴) → (𝑋 “ {1𝑜}) ∈ V)
4 cnveq 5266 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
54imaeq1d 5434 . . 3 (𝑥 = 𝑋 → (𝑥 “ {1𝑜}) = (𝑋 “ {1𝑜}))
6 pw2f1o2.f . . 3 𝐹 = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))
75, 6fvmptg 6247 . 2 ((𝑋 ∈ (2𝑜𝑚 𝐴) ∧ (𝑋 “ {1𝑜}) ∈ V) → (𝐹𝑋) = (𝑋 “ {1𝑜}))
83, 7mpdan 701 1 (𝑋 ∈ (2𝑜𝑚 𝐴) → (𝐹𝑋) = (𝑋 “ {1𝑜}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3190  {csn 4155  cmpt 4683  ccnv 5083  cima 5087  cfv 5857  (class class class)co 6615  1𝑜c1o 7513  2𝑜c2o 7514  𝑚 cmap 7817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fv 5865
This theorem is referenced by:  pw2f1o2val2  37126
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