Theorem f1opw 7701

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⟶1-1onto𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴⟶1-1onto𝒫 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 23	. 2 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → 𝐹:𝐴⟶1-1onto𝐵)
2		dff1o3 6853	. . 3 ⊢ (𝐹:𝐴⟶1-1onto𝐵 ↔ (𝐹:𝐴⟶onto𝐵 ∧ Fun ◡𝐹))
3		vex 3458	. . . 4 ⊢ 𝑎 ∈ V
4	3	funimaex 6649	. . 3 ⊢ (Fun ◡𝐹 → (◡𝐹 “ 𝑎) ∈ V)
5	2, 4	simplbiim 514	. 2 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → (◡𝐹 “ 𝑎) ∈ V)
6		f1ofun 6848	. . 3 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → Fun 𝐹)
7		vex 3458	. . . 4 ⊢ 𝑏 ∈ V
8	7	funimaex 6649	. . 3 ⊢ (Fun 𝐹 → (𝐹 “ 𝑏) ∈ V)
9	6, 8	syl 18	. 2 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → (𝐹 “ 𝑏) ∈ V)
10	1, 5, 9	f1opw2 7700	1 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴⟶1-1onto𝒫 𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Vcvv 3454  𝒫 cpw 4560   ↦ cmpt 5189  ^◡ccnv 5654   “ cima 5658  Fun wfun 6552  ⟶ontowfo 6556  ⟶1-1ontowf1o 6557

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565

This theorem is used by:  ackbij2lem2  10344