Theorem fopwdom 8608

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 5907	. . . . . 6 ⊢ (◡𝐹 “ 𝑎) ⊆ ran ◡𝐹
2		dfdm4 5728	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
3		fof 6565	. . . . . . . 8 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
4	3	fdmd 6497	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
5	2, 4	syl5eqr 2847	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran ◡𝐹 = 𝐴)
6	1, 5	sseqtrid 3967	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → (◡𝐹 “ 𝑎) ⊆ 𝐴)
7	6	adantl 485	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → (◡𝐹 “ 𝑎) ⊆ 𝐴)
8		cnvexg 7611	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → ◡𝐹 ∈ V)
9	8	adantr 484	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ◡𝐹 ∈ V)
10		imaexg 7602	. . . . 5 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ 𝑎) ∈ V)
11		elpwg 4500	. . . . 5 ⊢ ((◡𝐹 “ 𝑎) ∈ V → ((◡𝐹 “ 𝑎) ∈ 𝒫 𝐴 ↔ (◡𝐹 “ 𝑎) ⊆ 𝐴))
12	9, 10, 11	3syl 18	. . . 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 25	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → (𝑎 ∈ 𝒫 𝐵 → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴))
15		imaeq2 5892	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → (𝐹 “ (◡𝐹 “ 𝑎)) = (𝐹 “ (◡𝐹 “ 𝑏)))
16	15	adantl 485	. . . . . 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 775	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝐹:𝐴–onto→𝐵)
18		simplrl 776	. . . . . . . 8 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ∈ 𝒫 𝐵)
19	18	elpwid 4508	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ⊆ 𝐵)
20		foimacnv 6607	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑎 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
21	17, 19, 20	syl2anc 587	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
22		simplrr 777	. . . . . . . 8 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ∈ 𝒫 𝐵)
23	22	elpwid 4508	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ⊆ 𝐵)
24		foimacnv 6607	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑏 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑏)) = 𝑏)
25	17, 23, 24	syl2anc 587	. . . . . 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 2841	. . . . 5 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 = 𝑏)
27	26	ex 416	. . . 4 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → 𝑎 = 𝑏))
28		imaeq2 5892	. . . 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 416	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ((𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) ↔ 𝑎 = 𝑏)))
31		rnexg 7595	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V)
32		forn 6568	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
33	32	eleq1d 2874	. . . . 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 410	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ V)
36	35	pwexd 5245	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ∈ V)
37		dmfex 7623	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
38	3, 37	sylan2 595	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝐴 ∈ V)
39	38	pwexd 5245	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐴 ∈ V)
40	14, 30, 36, 39	dom3d 8534	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497   class class class wbr 5030  ^◡ccnv 5518  dom cdm 5519  ran crn 5520   “ cima 5522  ⟶wf 6320  –onto→wfo 6322   ≼ cdom 8490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-fv 6332  df-dom 8494

This theorem is referenced by:  pwdom  8653  wdompwdom  9026