Theorem fopwdom 6310

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 4679	. . . . . 6 ⊢ (◡𝐹 “ 𝑎) ⊆ ran ◡𝐹
2		dfdm4 4527	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
3		fof 5106	. . . . . . . 8 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
4		fdm 5050	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
5	3, 4	syl 14	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
6	2, 5	syl5eqr 2086	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran ◡𝐹 = 𝐴)
7	1, 6	syl5sseq 2993	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → (◡𝐹 “ 𝑎) ⊆ 𝐴)
8	7	adantl 262	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → (◡𝐹 “ 𝑎) ⊆ 𝐴)
9		cnvexg 4855	. . . . . 6 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
10	9	adantr 261	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → ◡𝐹 ∈ V)
11		imaexg 4680	. . . . 5 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ 𝑎) ∈ V)
12		elpwg 3367	. . . . 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 156	. . 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 4664	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → (𝐹 “ (◡𝐹 “ 𝑎)) = (𝐹 “ (◡𝐹 “ 𝑏)))
17	16	adantl 262	. . . . . 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 486	. . . . . . 7 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝐹:𝐴–onto→𝐵)
19		simplrl 487	. . . . . . . 8 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ∈ 𝒫 𝐵)
20	19	elpwid 3369	. . . . . . 7 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ⊆ 𝐵)
21		foimacnv 5144	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑎 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
22	18, 20, 21	syl2anc 391	. . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
23		simplrr 488	. . . . . . . 8 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ∈ 𝒫 𝐵)
24	23	elpwid 3369	. . . . . . 7 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ⊆ 𝐵)
25		foimacnv 5144	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑏 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑏)) = 𝑏)
26	18, 24, 25	syl2anc 391	. . . . . 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 2080	. . . . 5 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 = 𝑏)
28	27	ex 108	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → 𝑎 = 𝑏))
29		imaeq2 4664	. . . 4 ⊢ (𝑎 = 𝑏 → (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏))
30	28, 29	impbid1 130	. . 3 ⊢ (((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) ↔ 𝑎 = 𝑏))
31	30	ex 108	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → ((𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) ↔ 𝑎 = 𝑏)))
32		rnexg 4597	. . . . 5 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V)
33		forn 5109	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
34	33	eleq1d 2106	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
35	32, 34	syl5ibcom 144	. . . 4 ⊢ (𝐹 ∈ V → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V))
36	35	imp 115	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ V)
37		pwexg 3933	. . 3 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
38	36, 37	syl 14	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ∈ V)
39		dmfex 5079	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
40	3, 39	sylan2 270	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝐴 ∈ V)
41		pwexg 3933	. . 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 6254	1 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  𝒫 cpw 3359   class class class wbr 3764  ^◡ccnv 4344  dom cdm 4345  ran crn 4346   “ cima 4348  ⟶wf 4898  –onto→wfo 4900   ≼ cdom 6220

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-fv 4910  df-dom 6223

This theorem is referenced by: (None)