Theorem pw2f1o2val 43032

Description: Function value of the pw2f1o2 43031 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
pw2f1o2val	⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o}))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem pw2f1o2val

Step	Hyp	Ref	Expression
1		cnvexg 7857	. . 3 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → ◡𝑋 ∈ V)
2		imaexg 7846	. . 3 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ {1o}) ∈ V)
3	1, 2	syl 17	. 2 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (◡𝑋 “ {1o}) ∈ V)
4		cnveq 5816	. . . 4 ⊢ (𝑥 = 𝑋 → ◡𝑥 = ◡𝑋)
5	4	imaeq1d 6010	. . 3 ⊢ (𝑥 = 𝑋 → (◡𝑥 “ {1o}) = (◡𝑋 “ {1o}))
6		pw2f1o2.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o}))
7	5, 6	fvmptg 6928	. 2 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ (◡𝑋 “ {1o}) ∈ V) → (𝐹‘𝑋) = (◡𝑋 “ {1o}))
8	3, 7	mpdan 687	1 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o}))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3436  {csn 4577   ↦ cmpt 5173  ^◡ccnv 5618   “ cima 5622  ‘cfv 6482  (class class class)co 7349  1_oc1o 8381  2_oc2o 8382   ↑_m cmap 8753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fv 6490

This theorem is referenced by:  pw2f1o2val2  43033