Theorem f1opw 6152

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 5527	. . . 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 2774	. . . 4 ⊢ 𝑎 ∈ V
5	4	funimaex 5358	. . 3 ⊢ (Fun ◡𝐹 → (◡𝐹 “ 𝑎) ∈ V)
6	3, 5	syl 14	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 “ 𝑎) ∈ V)
7		f1ofun 5523	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)
8		vex 2774	. . . 4 ⊢ 𝑏 ∈ V
9	8	funimaex 5358	. . 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 6151	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2175  Vcvv 2771  𝒫 cpw 3615   ↦ cmpt 4104  ^◡ccnv 4673   “ cima 4677  Fun wfun 5264  –onto→wfo 5268  –1-1-onto→wf1o 5269

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277

This theorem is referenced by: (None)