Theorem pw2f1o2val2 43360

Description: Membership in a mapped set under the pw2f1o2 43358 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Hypothesis

Ref	Expression
pw2f1o2.f	⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o}))

Assertion

Ref	Expression
pw2f1o2val2	⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1o))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋   𝑥,𝑌

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem pw2f1o2val2

Step	Hyp	Ref	Expression
1		pw2f1o2.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o}))
2	1	pw2f1o2val 43359	. . . 4 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o}))
3	2	eleq2d 2823	. . 3 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ (◡𝑋 “ {1o})))
4	3	adantr 480	. 2 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ (◡𝑋 “ {1o})))
5		elmapi 8791	. . . 4 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → 𝑋:𝐴⟶2o)
6		ffn 6663	. . . 4 ⊢ (𝑋:𝐴⟶2o → 𝑋 Fn 𝐴)
7		fniniseg 7007	. . . 4 ⊢ (𝑋 Fn 𝐴 → (𝑌 ∈ (◡𝑋 “ {1o}) ↔ (𝑌 ∈ 𝐴 ∧ (𝑋‘𝑌) = 1o)))
8	5, 6, 7	3syl 18	. . 3 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝑌 ∈ (◡𝑋 “ {1o}) ↔ (𝑌 ∈ 𝐴 ∧ (𝑋‘𝑌) = 1o)))
9	8	baibd 539	. 2 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (◡𝑋 “ {1o}) ↔ (𝑋‘𝑌) = 1o))
10	4, 9	bitrd 279	1 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1o))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {csn 4581   ↦ cmpt 5180  ^◡ccnv 5624   “ cima 5628   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  1_oc1o 8393  2_oc2o 8394   ↑_m cmap 8768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-map 8770

This theorem is referenced by:  wepwsolem  43362