Theorem fopwdom 9142

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⟶onto𝐵) → 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 6216	. . . . . 6 ⊢ (◡𝐹 “ 𝑎) ⊆ ran ◡𝐹
2		dfdm4 5889	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
3		fof 6826	. . . . . . . 8 ⊢ (𝐹:𝐴⟶onto𝐵 → 𝐹:𝐴⟶𝐵)
4	3	fdmd 6750	. . . . . . 7 ⊢ (𝐹:𝐴⟶onto𝐵 → dom 𝐹 = 𝐴)
5	2, 4	eqtr3id 2812	. . . . . 6 ⊢ (𝐹:𝐴⟶onto𝐵 → ran ◡𝐹 = 𝐴)
6	1, 5	sseqtrid 3955	. . . . 5 ⊢ (𝐹:𝐴⟶onto𝐵 → (◡𝐹 “ 𝑎) ⊆ 𝐴)
7	6	adantl 487	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) → (◡𝐹 “ 𝑎) ⊆ 𝐴)
8		cnvexg 7970	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → ◡𝐹 ∈ V)
9	8	adantr 486	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) → ◡𝐹 ∈ V)
10		imaexg 7959	. . . . 5 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ 𝑎) ∈ V)
11		elpwg 4563	. . . . 5 ⊢ ((◡𝐹 “ 𝑎) ∈ V → ((◡𝐹 “ 𝑎) ∈ 𝒫 𝐴 ↔ (◡𝐹 “ 𝑎) ⊆ 𝐴))
12	9, 10, 11	3syl 19	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) → ((◡𝐹 “ 𝑎) ∈ 𝒫 𝐴 ↔ (◡𝐹 “ 𝑎) ⊆ 𝐴))
13	7, 12	mpbird 260	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴)
14	13	a1d 26	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) → (𝑎 ∈ 𝒫 𝐵 → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴))
15		imaeq2 6066	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → (𝐹 “ (◡𝐹 “ 𝑎)) = (𝐹 “ (◡𝐹 “ 𝑏)))
16	15	adantl 487	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → (𝐹 “ (◡𝐹 “ 𝑎)) = (𝐹 “ (◡𝐹 “ 𝑏)))
17		simpllr 788	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝐹:𝐴⟶onto𝐵)
18		simplrl 789	. . . . . . . 8 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ∈ 𝒫 𝐵)
19	18	elpwid 4569	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 ⊆ 𝐵)
20		foimacnv 6872	. . . . . . 7 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝑎 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
21	17, 19, 20	syl2anc 596	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
22		simplrr 790	. . . . . . . 8 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ∈ 𝒫 𝐵)
23	22	elpwid 4569	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑏 ⊆ 𝐵)
24		foimacnv 6872	. . . . . . 7 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝑏 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑏)) = 𝑏)
25	17, 23, 24	syl2anc 596	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → (𝐹 “ (◡𝐹 “ 𝑏)) = 𝑏)
26	16, 21, 25	3eqtr3d 2806	. . . . 5 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) ∧ (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏)) → 𝑎 = 𝑏)
27	26	ex 418	. . . 4 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) → 𝑎 = 𝑏))
28		imaeq2 6066	. . . 4 ⊢ (𝑎 = 𝑏 → (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏))
29	27, 28	impbid1 228	. . 3 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) ∧ (𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵)) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) ↔ 𝑎 = 𝑏))
30	29	ex 418	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) → ((𝑎 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ 𝒫 𝐵) → ((◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑏) ↔ 𝑎 = 𝑏)))
31		rnexg 7948	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V)
32		forn 6829	. . . . . 6 ⊢ (𝐹:𝐴⟶onto𝐵 → ran 𝐹 = 𝐵)
33	32	eleq1d 2849	. . . . 5 ⊢ (𝐹:𝐴⟶onto𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
34	31, 33	syl5ibcom 248	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴⟶onto𝐵 → 𝐵 ∈ V))
35	34	imp 412	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) → 𝐵 ∈ V)
36	35	pwexd 5348	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) → 𝒫 𝐵 ∈ V)
37		dmfex 7951	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
38	3, 37	sylan2 605	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) → 𝐴 ∈ V)
39	38	pwexd 5348	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) → 𝒫 𝐴 ∈ V)
40	14, 30, 36, 39	dom3d 9059	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560   class class class wbr 5108  ^◡ccnv 5657  dom cdm 5658  ran crn 5659   “ cima 5661  ⟶wf 6562  ⟶ontowfo 6564   ≼ cdom 9009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-fv 6574  df-dom 9013

This theorem is used by:  pwdom  9186  wdompwdom  9609