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Theorem pw2f1o2val2 41407
Description: Membership in a mapped set under the pw2f1o2 41405 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 41406 . . . 4 (𝑋 ∈ (2om 𝐴) → (𝐹𝑋) = (𝑋 “ {1o}))
32eleq2d 2820 . . 3 (𝑋 ∈ (2om 𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ (𝑋 “ {1o})))
43adantr 482 . 2 ((𝑋 ∈ (2om 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ (𝑋 “ {1o})))
5 elmapi 8790 . . . 4 (𝑋 ∈ (2om 𝐴) → 𝑋:𝐴⟶2o)
6 ffn 6669 . . . 4 (𝑋:𝐴⟶2o𝑋 Fn 𝐴)
7 fniniseg 7011 . . . 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 541 . 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 205  wa 397   = wceq 1542  wcel 2107  {csn 4587  cmpt 5189  ccnv 5633  cima 5637   Fn wfn 6492  wf 6493  cfv 6497  (class class class)co 7358  1oc1o 8406  2oc2o 8407  m cmap 8768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770
This theorem is referenced by:  wepwsolem  41412
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