Theorem fopwdom 6814

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 4964	. . . . . 6 ⊢ (◡𝐹 “ 𝑎) ⊆ ran ◡𝐹
2		dfdm4 4803	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
3		fof 5420	. . . . . . . 8 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
4		fdm 5353	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
5	3, 4	syl 14	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
6	2, 5	eqtr3id 2217	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran ◡𝐹 = 𝐴)
7	1, 6	sseqtrid 3197	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → (◡𝐹 “ 𝑎) ⊆ 𝐴)
8	7	adantl 275	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → (◡𝐹 “ 𝑎) ⊆ 𝐴)
9		cnvexg 5148	. . . . . 6 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
10	9	adantr 274	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → ◡𝐹 ∈ V)
11		imaexg 4965	. . . . 5 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ 𝑎) ∈ V)
12		elpwg 3574	. . . . 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 166	. . 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 4949	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → (𝐹 “ (◡𝐹 “ 𝑎)) = (𝐹 “ (◡𝐹 “ 𝑏)))
17	16	adantl 275	. . . . . 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 529	. . . . . . 7 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝐹:𝐴–onto→𝐵)
19		simplrl 530	. . . . . . . 8 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ∈ 𝒫 𝐵)
20	19	elpwid 3577	. . . . . . 7 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ⊆ 𝐵)
21		foimacnv 5460	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑎 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
22	18, 20, 21	syl2anc 409	. . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
23		simplrr 531	. . . . . . . 8 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ∈ 𝒫 𝐵)
24	23	elpwid 3577	. . . . . . 7 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ⊆ 𝐵)
25		foimacnv 5460	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑏 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑏)) = 𝑏)
26	18, 24, 25	syl2anc 409	. . . . . 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 2211	. . . . 5 ⊢ ((((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 = 𝑏)
28	27	ex 114	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → 𝑎 = 𝑏))
29		imaeq2 4949	. . . 4 ⊢ (𝑎 = 𝑏 → (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏))
30	28, 29	impbid1 141	. . 3 ⊢ (((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) ↔ 𝑎 = 𝑏))
31	30	ex 114	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → ((𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) ↔ 𝑎 = 𝑏)))
32		rnexg 4876	. . . . 5 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V)
33		forn 5423	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
34	33	eleq1d 2239	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
35	32, 34	syl5ibcom 154	. . . 4 ⊢ (𝐹 ∈ V → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V))
36	35	imp 123	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ V)
37		pwexg 4166	. . 3 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
38	36, 37	syl 14	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ∈ V)
39		dmfex 5387	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
40	3, 39	sylan2 284	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝐴 ∈ V)
41		pwexg 4166	. . 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 6752	1 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  Vcvv 2730   ⊆ wss 3121  𝒫 cpw 3566   class class class wbr 3989  ^◡ccnv 4610  dom cdm 4611  ran crn 4612   “ cima 4614  ⟶wf 5194  –onto→wfo 5196   ≼ cdom 6717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-fv 5206  df-dom 6720

This theorem is referenced by: (None)