Theorem pw2f1o2val 44046

Description: Function value of the pw2f1o2 44045 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 7936	. . 3 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → ◡𝑋 ∈ V)
2		imaexg 7925	. . 3 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ {1o}) ∈ V)
3	1, 2	syl 18	. 2 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (◡𝑋 “ {1o}) ∈ V)
4		cnveq 5850	. . . 4 ⊢ (𝑥 = 𝑋 → ◡𝑥 = ◡𝑋)
5	4	imaeq1d 6050	. . 3 ⊢ (𝑥 = 𝑋 → (◡𝑥 “ {1o}) = (◡𝑋 “ {1o}))
6		pw2f1o2.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o}))
7	5, 6	fvmptg 6991	. 2 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ (◡𝑋 “ {1o}) ∈ V) → (𝐹‘𝑋) = (◡𝑋 “ {1o}))
8	3, 7	mpdan 700	1 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o}))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  {csn 4583   ↦ cmpt 5185  ^◡ccnv 5649   “ cima 5653  ‘cfv 6538  (class class class)co 7420  1_oc1o 8469  2_oc2o 8470   ↑_m cmap 8847

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fv 6546

This theorem is used by:  pw2f1o2val2  44047