Theorem fopwdom 6836

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

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 4982	. . . . . 6 ⊢ (◡𝐹 “ 𝑎) ⊆ ran ◡𝐹
2		dfdm4 4820	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
3		fof 5439	. . . . . . . 8 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
4		fdm 5372	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
5	3, 4	syl 14	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
6	2, 5	eqtr3id 2224	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran ◡𝐹 = 𝐴)
7	1, 6	sseqtrid 3206	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → (◡𝐹 “ 𝑎) ⊆ 𝐴)
8	7	adantl 277	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → (◡𝐹 “ 𝑎) ⊆ 𝐴)
9		cnvexg 5167	. . . . . 6 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
10	9	adantr 276	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → ◡𝐹 ∈ V)
11		imaexg 4983	. . . . 5 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ 𝑎) ∈ V)
12		elpwg 3584	. . . . 5 ⊢ ((◡𝐹 “ 𝑎) ∈ V → ((◡𝐹 “ 𝑎) ∈ 𝒫 𝐴 ↔ (◡𝐹 “ 𝑎) ⊆ 𝐴))
13	10, 11, 12	3syl 17	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → ((◡𝐹 “ 𝑎) ∈ 𝒫 𝐴 ↔ (◡𝐹 “ 𝑎) ⊆ 𝐴))
14	8, 13	mpbird 167	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴)
15	14	a1d 22	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → (𝑎 ∈ 𝒫 𝐵 → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴))
16		imaeq2 4967	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → (𝐹 “ (◡𝐹 “ 𝑎)) = (𝐹 “ (◡𝐹 “ 𝑏)))
17	16	adantl 277	. . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → (𝐹 “ (◡𝐹 “ 𝑎)) = (𝐹 “ (◡𝐹 “ 𝑏)))
18		simpllr 534	. . . . . . 7 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝐹:𝐴–onto→𝐵)
19		simplrl 535	. . . . . . . 8 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ∈ 𝒫 𝐵)
20	19	elpwid 3587	. . . . . . 7 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ⊆ 𝐵)
21		foimacnv 5480	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑎 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
22	18, 20, 21	syl2anc 411	. . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
23		simplrr 536	. . . . . . . 8 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ∈ 𝒫 𝐵)
24	23	elpwid 3587	. . . . . . 7 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ⊆ 𝐵)
25		foimacnv 5480	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑏 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑏)) = 𝑏)
26	18, 24, 25	syl2anc 411	. . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → (𝐹 “ (◡𝐹 “ 𝑏)) = 𝑏)
27	17, 22, 26	3eqtr3d 2218	. . . . 5 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 = 𝑏)
28	27	ex 115	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → 𝑎 = 𝑏))
29		imaeq2 4967	. . . 4 ⊢ (𝑎 = 𝑏 → (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏))
30	28, 29	impbid1 142	. . 3 ⊢ (((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) ↔ 𝑎 = 𝑏))
31	30	ex 115	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → ((𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) ↔ 𝑎 = 𝑏)))
32		rnexg 4893	. . . . 5 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V)
33		forn 5442	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
34	33	eleq1d 2246	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
35	32, 34	syl5ibcom 155	. . . 4 ⊢ (𝐹 ∈ V → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V))
36	35	imp 124	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ V)
37		pwexg 4181	. . 3 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
38	36, 37	syl 14	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ∈ V)
39		dmfex 5406	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
40	3, 39	sylan2 286	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝐴 ∈ V)
41		pwexg 4181	. . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
42	40, 41	syl 14	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐴 ∈ V)
43	15, 31, 38, 42	dom3d 6774	1 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  Vcvv 2738   ⊆ wss 3130  𝒫 cpw 3576   class class class wbr 4004  ^◡ccnv 4626  dom cdm 4627  ran crn 4628   “ cima 4630  ⟶wf 5213  –onto→wfo 5215   ≼ cdom 6739

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-fv 5225  df-dom 6742

This theorem is referenced by: (None)