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Theorem pw2f1o2val2 38387
Description: Membership in a mapped set under the pw2f1o2 38385 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 38386 . . . 4 (𝑋 ∈ (2𝑜𝑚 𝐴) → (𝐹𝑋) = (𝑋 “ {1𝑜}))
32eleq2d 2865 . . 3 (𝑋 ∈ (2𝑜𝑚 𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ (𝑋 “ {1𝑜})))
43adantr 473 . 2 ((𝑋 ∈ (2𝑜𝑚 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ (𝑋 “ {1𝑜})))
5 elmapi 8118 . . . 4 (𝑋 ∈ (2𝑜𝑚 𝐴) → 𝑋:𝐴⟶2𝑜)
6 ffn 6257 . . . 4 (𝑋:𝐴⟶2𝑜𝑋 Fn 𝐴)
7 fniniseg 6565 . . . 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 536 . 2 ((𝑋 ∈ (2𝑜𝑚 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝑋 “ {1𝑜}) ↔ (𝑋𝑌) = 1𝑜))
104, 9bitrd 271 1 ((𝑋 ∈ (2𝑜𝑚 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝑌) = 1𝑜))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  {csn 4369  cmpt 4923  ccnv 5312  cima 5316   Fn wfn 6097  wf 6098  cfv 6102  (class class class)co 6879  1𝑜c1o 7793  2𝑜c2o 7794  𝑚 cmap 8096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-1st 7402  df-2nd 7403  df-map 8098
This theorem is referenced by:  wepwsolem  38392
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