Theorem fopwdom 9072

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 6074	. . . . . 6 ⊢ (◡𝐹 “ 𝑎) ⊆ ran ◡𝐹
2		dfdm4 5886	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
3		fof 6793	. . . . . . . 8 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
4	3	fdmd 6717	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
5	2, 4	eqtr3id 2818	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran ◡𝐹 = 𝐴)
6	1, 5	sseqtrid 3987	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → (◡𝐹 “ 𝑎) ⊆ 𝐴)
7	6	adantl 486	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → (◡𝐹 “ 𝑎) ⊆ 𝐴)
8		cnvexg 7920	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → ◡𝐹 ∈ V)
9	8	adantr 485	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ◡𝐹 ∈ V)
10		imaexg 7909	. . . . 5 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ 𝑎) ∈ V)
11		elpwg 4570	. . . . 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 6059	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → (𝐹 “ (◡𝐹 “ 𝑎)) = (𝐹 “ (◡𝐹 “ 𝑏)))
16	15	adantl 486	. . . . . 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 787	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝐹:𝐴–onto→𝐵)
18		simplrl 788	. . . . . . . 8 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ∈ 𝒫 𝐵)
19	18	elpwid 4576	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ⊆ 𝐵)
20		foimacnv 6839	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑎 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
21	17, 19, 20	syl2anc 595	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
22		simplrr 789	. . . . . . . 8 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ∈ 𝒫 𝐵)
23	22	elpwid 4576	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ⊆ 𝐵)
24		foimacnv 6839	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑏 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑏)) = 𝑏)
25	17, 23, 24	syl2anc 595	. . . . . 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 2812	. . . . 5 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 = 𝑏)
27	26	ex 417	. . . 4 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → 𝑎 = 𝑏))
28		imaeq2 6059	. . . 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 417	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ((𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) ↔ 𝑎 = 𝑏)))
31		rnexg 7898	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V)
32		forn 6796	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
33	32	eleq1d 2854	. . . . 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 411	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ V)
36	35	pwexd 5351	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ∈ V)
37		dmfex 7901	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
38	3, 37	sylan2 604	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝐴 ∈ V)
39	38	pwexd 5351	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐴 ∈ V)
40	14, 30, 36, 39	dom3d 8990	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913  𝒫 cpw 4567   class class class wbr 5113  ^◡ccnv 5661  dom cdm 5662  ran crn 5663   “ cima 5665  ⟶wf 6533  –onto→wfo 6535   ≼ cdom 8940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-fv 6545  df-dom 8944

This theorem is referenced by:  pwdom  9116  wdompwdom  9539