Theorem f1opw 6262

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 19	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1-onto→𝐵)
2		dff1o3 5620	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹))
3	2	simprbi 275	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun ◡𝐹)
4		vex 2816	. . . 4 ⊢ 𝑎 ∈ V
5	4	funimaex 5441	. . 3 ⊢ (Fun ◡𝐹 → (◡𝐹 “ 𝑎) ∈ V)
6	3, 5	syl 14	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 “ 𝑎) ∈ V)
7		f1ofun 5616	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)
8		vex 2816	. . . 4 ⊢ 𝑏 ∈ V
9	8	funimaex 5441	. . 3 ⊢ (Fun 𝐹 → (𝐹 “ 𝑏) ∈ V)
10	7, 9	syl 14	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 “ 𝑏) ∈ V)
11	1, 6, 10	f1opw2 6261	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2203  Vcvv 2813  𝒫 cpw 3669   ↦ cmpt 4171  ^◡ccnv 4748   “ cima 4752  Fun wfun 5346  –onto→wfo 5350  –1-1-onto→wf1o 5351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359

This theorem is referenced by: (None)