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Theorem pw2f1o2val2 36625
 Description: Membership in a mapped set under the pw2f1o2 36623 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 36624 . . . 4 (𝑋 ∈ (2𝑜𝑚 𝐴) → (𝐹𝑋) = (𝑋 “ {1𝑜}))
32eleq2d 2673 . . 3 (𝑋 ∈ (2𝑜𝑚 𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ (𝑋 “ {1𝑜})))
43adantr 480 . 2 ((𝑋 ∈ (2𝑜𝑚 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ (𝑋 “ {1𝑜})))
5 elmapi 7765 . . . 4 (𝑋 ∈ (2𝑜𝑚 𝐴) → 𝑋:𝐴⟶2𝑜)
6 ffn 5958 . . . 4 (𝑋:𝐴⟶2𝑜𝑋 Fn 𝐴)
7 fniniseg 6246 . . . 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 946 . 2 ((𝑋 ∈ (2𝑜𝑚 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝑋 “ {1𝑜}) ↔ (𝑋𝑌) = 1𝑜))
104, 9bitrd 267 1 ((𝑋 ∈ (2𝑜𝑚 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝑌) = 1𝑜))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125   ↦ cmpt 4643  ◡ccnv 5037   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1𝑜c1o 7440  2𝑜c2o 7441   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by:  wepwsolem  36630
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