Theorem pw2f1o2val2 42999

Description: Membership in a mapped set under the pw2f1o2 42997 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 42998	. . . 4 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o}))
3	2	eleq2d 2830	. . 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 8909	. . . 4 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → 𝑋:𝐴⟶2o)
6		ffn 6749	. . . 4 ⊢ (𝑋:𝐴⟶2o → 𝑋 Fn 𝐴)
7		fniniseg 7095	. . . 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 1537   ∈ wcel 2108  {csn 4648   ↦ cmpt 5249  ^◡ccnv 5699   “ cima 5703   Fn wfn 6570  ⟶wf 6571  ‘cfv 6575  (class class class)co 7450  1_oc1o 8517  2_oc2o 8518   ↑_m cmap 8886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-1st 8032  df-2nd 8033  df-map 8888

This theorem is referenced by:  wepwsolem  43001