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Theorem pw2f1o2val2 42999
Description: Membership in a mapped set under the pw2f1o2 42997 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
pw2f1o2val2 ((𝑋 ∈ (2om 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝑌) = 1o))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋   𝑥,𝑌
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem pw2f1o2val2
StepHypRef Expression
1 pw2f1o2.f . . . . 5 𝐹 = (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o}))
21pw2f1o2val 42998 . . . 4 (𝑋 ∈ (2om 𝐴) → (𝐹𝑋) = (𝑋 “ {1o}))
32eleq2d 2830 . . 3 (𝑋 ∈ (2om 𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ (𝑋 “ {1o})))
43adantr 480 . 2 ((𝑋 ∈ (2om 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ (𝑋 “ {1o})))
5 elmapi 8909 . . . 4 (𝑋 ∈ (2om 𝐴) → 𝑋:𝐴⟶2o)
6 ffn 6749 . . . 4 (𝑋:𝐴⟶2o𝑋 Fn 𝐴)
7 fniniseg 7095 . . . 4 (𝑋 Fn 𝐴 → (𝑌 ∈ (𝑋 “ {1o}) ↔ (𝑌𝐴 ∧ (𝑋𝑌) = 1o)))
85, 6, 73syl 18 . . 3 (𝑋 ∈ (2om 𝐴) → (𝑌 ∈ (𝑋 “ {1o}) ↔ (𝑌𝐴 ∧ (𝑋𝑌) = 1o)))
98baibd 539 . 2 ((𝑋 ∈ (2om 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝑋 “ {1o}) ↔ (𝑋𝑌) = 1o))
104, 9bitrd 279 1 ((𝑋 ∈ (2om 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝑌) = 1o))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2108  {csn 4648  cmpt 5249  ccnv 5699  cima 5703   Fn wfn 6570  wf 6571  cfv 6575  (class class class)co 7450  1oc1o 8517  2oc2o 8518  m cmap 8886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-1st 8032  df-2nd 8033  df-map 8888
This theorem is referenced by:  wepwsolem  43001
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