Theorem fopwdom 9080

Description: Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) (Revised by AV, 18-Sep-2021.)

Assertion

Ref	Expression
fopwdom	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)

Proof of Theorem fopwdom

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imassrn 6075	. . . . . 6 ⊢ (◡𝐹 “ 𝑎) ⊆ ran ◡𝐹
2		dfdm4 5887	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
3		fof 6796	. . . . . . . 8 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
4	3	fdmd 6720	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
5	2, 4	eqtr3id 2814	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran ◡𝐹 = 𝐴)
6	1, 5	sseqtrid 3980	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → (◡𝐹 “ 𝑎) ⊆ 𝐴)
7	6	adantl 487	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → (◡𝐹 “ 𝑎) ⊆ 𝐴)
8		cnvexg 7927	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → ◡𝐹 ∈ V)
9	8	adantr 486	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ◡𝐹 ∈ V)
10		imaexg 7916	. . . . 5 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ 𝑎) ∈ V)
11		elpwg 4567	. . . . 5 ⊢ ((◡𝐹 “ 𝑎) ∈ V → ((◡𝐹 “ 𝑎) ∈ 𝒫 𝐴 ↔ (◡𝐹 “ 𝑎) ⊆ 𝐴))
12	9, 10, 11	3syl 19	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ((◡𝐹 “ 𝑎) ∈ 𝒫 𝐴 ↔ (◡𝐹 “ 𝑎) ⊆ 𝐴))
13	7, 12	mpbird 260	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴)
14	13	a1d 26	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → (𝑎 ∈ 𝒫 𝐵 → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴))
15		imaeq2 6060	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → (𝐹 “ (◡𝐹 “ 𝑎)) = (𝐹 “ (◡𝐹 “ 𝑏)))
16	15	adantl 487	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → (𝐹 “ (◡𝐹 “ 𝑎)) = (𝐹 “ (◡𝐹 “ 𝑏)))
17		simpllr 788	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝐹:𝐴–onto→𝐵)
18		simplrl 789	. . . . . . . 8 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ∈ 𝒫 𝐵)
19	18	elpwid 4573	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ⊆ 𝐵)
20		foimacnv 6842	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑎 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
21	17, 19, 20	syl2anc 596	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
22		simplrr 790	. . . . . . . 8 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ∈ 𝒫 𝐵)
23	22	elpwid 4573	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ⊆ 𝐵)
24		foimacnv 6842	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑏 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑏)) = 𝑏)
25	17, 23, 24	syl2anc 596	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → (𝐹 “ (◡𝐹 “ 𝑏)) = 𝑏)
26	16, 21, 25	3eqtr3d 2808	. . . . 5 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 = 𝑏)
27	26	ex 418	. . . 4 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → 𝑎 = 𝑏))
28		imaeq2 6060	. . . 4 ⊢ (𝑎 = 𝑏 → (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏))
29	27, 28	impbid1 228	. . 3 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) ↔ 𝑎 = 𝑏))
30	29	ex 418	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ((𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) ↔ 𝑎 = 𝑏)))
31		rnexg 7905	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V)
32		forn 6799	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
33	32	eleq1d 2850	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
34	31, 33	syl5ibcom 248	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V))
35	34	imp 412	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ V)
36	35	pwexd 5352	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ∈ V)
37		dmfex 7908	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
38	3, 37	sylan2 605	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝐴 ∈ V)
39	38	pwexd 5352	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐴 ∈ V)
40	14, 30, 36, 39	dom3d 8997	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ⊆ wss 3906  𝒫 cpw 4564   class class class wbr 5111  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   “ cima 5666  ⟶wf 6536  –onto→wfo 6538   ≼ cdom 8947

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-fv 6548  df-dom 8951

This theorem is used by:  pwdom  9124  wdompwdom  9547