Theorem f1opw 7647

Description: A one-to-one mapping induces a one-to-one mapping on power sets. (Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
f1opw	⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵)

Distinct variable groups:   𝐴,𝑏   𝐵,𝑏   𝐹,𝑏

Proof of Theorem f1opw

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1-onto→𝐵)
2		dff1o3 6808	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹))
3		vex 3454	. . . 4 ⊢ 𝑎 ∈ V
4	3	funimaex 6607	. . 3 ⊢ (Fun ◡𝐹 → (◡𝐹 “ 𝑎) ∈ V)
5	2, 4	simplbiim 504	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 “ 𝑎) ∈ V)
6		f1ofun 6804	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)
7		vex 3454	. . . 4 ⊢ 𝑏 ∈ V
8	7	funimaex 6607	. . 3 ⊢ (Fun 𝐹 → (𝐹 “ 𝑏) ∈ V)
9	6, 8	syl 17	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 “ 𝑏) ∈ V)
10	1, 5, 9	f1opw2 7646	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3450  𝒫 cpw 4565   ↦ cmpt 5190  ^◡ccnv 5639   “ cima 5643  Fun wfun 6507  –onto→wfo 6511  –1-1-onto→wf1o 6512

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pr 5389

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520

This theorem is referenced by:  ackbij2lem2  10198