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Theorem pw2f1o2val2 39630
Description: Membership in a mapped set under the pw2f1o2 39628 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 39629 . . . 4 (𝑋 ∈ (2om 𝐴) → (𝐹𝑋) = (𝑋 “ {1o}))
32eleq2d 2898 . . 3 (𝑋 ∈ (2om 𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ (𝑋 “ {1o})))
43adantr 483 . 2 ((𝑋 ∈ (2om 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ (𝑋 “ {1o})))
5 elmapi 8422 . . . 4 (𝑋 ∈ (2om 𝐴) → 𝑋:𝐴⟶2o)
6 ffn 6508 . . . 4 (𝑋:𝐴⟶2o𝑋 Fn 𝐴)
7 fniniseg 6824 . . . 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 542 . 2 ((𝑋 ∈ (2om 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝑋 “ {1o}) ↔ (𝑋𝑌) = 1o))
104, 9bitrd 281 1 ((𝑋 ∈ (2om 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝑌) = 1o))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  {csn 4560  cmpt 5138  ccnv 5548  cima 5552   Fn wfn 6344  wf 6345  cfv 6349  (class class class)co 7150  1oc1o 8089  2oc2o 8090  m cmap 8400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-map 8402
This theorem is referenced by:  wepwsolem  39635
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