Theorem pw2f1o2val 36624

Description: Function value of 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 ∈ (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 7005	. . 3 ⊢ (𝑋 ∈ (2𝑜 ↑𝑚 𝐴) → ◡𝑋 ∈ V)
2		imaexg 6995	. . 3 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ {1𝑜}) ∈ V)
3	1, 2	syl 17	. 2 ⊢ (𝑋 ∈ (2𝑜 ↑𝑚 𝐴) → (◡𝑋 “ {1𝑜}) ∈ V)
4		cnveq 5218	. . . 4 ⊢ (𝑥 = 𝑋 → ◡𝑥 = ◡𝑋)
5	4	imaeq1d 5384	. . 3 ⊢ (𝑥 = 𝑋 → (◡𝑥 “ {1𝑜}) = (◡𝑋 “ {1𝑜}))
6		pw2f1o2.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜}))
7	5, 6	fvmptg 6189	. 2 ⊢ ((𝑋 ∈ (2𝑜 ↑𝑚 𝐴) ∧ (◡𝑋 “ {1𝑜}) ∈ V) → (𝐹‘𝑋) = (◡𝑋 “ {1𝑜}))
8	3, 7	mpdan 699	1 ⊢ (𝑋 ∈ (2𝑜 ↑𝑚 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1𝑜}))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125   ↦ cmpt 4643  ^◡ccnv 5037   “ cima 5041  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440  2_𝑜c2o 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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  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-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-fv 5812

This theorem is referenced by:  pw2f1o2val2  36625