Theorem aceq3lem 8943

Description: Lemma for dfac3 8944. (Contributed by NM, 2-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypothesis

Ref	Expression
aceq3lem.1	⊢ 𝐹 = (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))

Assertion

Ref	Expression
aceq3lem	⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑢,𝑓

Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧,𝑤,𝑢,𝑓)

Proof of Theorem aceq3lem

Dummy variables ℎ 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . 6 ⊢ 𝑦 ∈ V
2	1	rnex 7100	. . . . 5 ⊢ ran 𝑦 ∈ V
3	2	pwex 4848	. . . 4 ⊢ 𝒫 ran 𝑦 ∈ V
4		raleq 3138	. . . . 5 ⊢ (𝑥 = 𝒫 ran 𝑦 → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
5	4	exbidv 1850	. . . 4 ⊢ (𝑥 = 𝒫 ran 𝑦 → (∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∃𝑓∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
6	3, 5	spcv 3299	. . 3 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑓∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
7		aceq3lem.1	. . . . . . 7 ⊢ 𝐹 = (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))
8		df-mpt 4730	. . . . . . 7 ⊢ (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢})) = {⟨𝑤, ℎ⟩ ∣ (𝑤 ∈ dom 𝑦 ∧ ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))}
9	7, 8	eqtri 2644	. . . . . 6 ⊢ 𝐹 = {⟨𝑤, ℎ⟩ ∣ (𝑤 ∈ dom 𝑦 ∧ ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))}
10		vex 3203	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
11	10	eldm 5321	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ dom 𝑦 ↔ ∃𝑢 𝑤𝑦𝑢)
12		abn0 3954	. . . . . . . . . . . . . 14 ⊢ ({𝑢 ∣ 𝑤𝑦𝑢} ≠ ∅ ↔ ∃𝑢 𝑤𝑦𝑢)
13	11, 12	bitr4i 267	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ dom 𝑦 ↔ {𝑢 ∣ 𝑤𝑦𝑢} ≠ ∅)
14		vex 3203	. . . . . . . . . . . . . . . . 17 ⊢ 𝑢 ∈ V
15	10, 14	brelrn 5356	. . . . . . . . . . . . . . . 16 ⊢ (𝑤𝑦𝑢 → 𝑢 ∈ ran 𝑦)
16	15	abssi 3677	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ 𝑤𝑦𝑢} ⊆ ran 𝑦
17	2	elpw2 4828	. . . . . . . . . . . . . . 15 ⊢ ({𝑢 ∣ 𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦 ↔ {𝑢 ∣ 𝑤𝑦𝑢} ⊆ ran 𝑦)
18	16, 17	mpbir 221	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∣ 𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦
19		neeq1 2856	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = {𝑢 ∣ 𝑤𝑦𝑢} → (𝑧 ≠ ∅ ↔ {𝑢 ∣ 𝑤𝑦𝑢} ≠ ∅))
20		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = {𝑢 ∣ 𝑤𝑦𝑢} → (𝑓‘𝑧) = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))
21		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = {𝑢 ∣ 𝑤𝑦𝑢} → 𝑧 = {𝑢 ∣ 𝑤𝑦𝑢})
22	20, 21	eleq12d 2695	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = {𝑢 ∣ 𝑤𝑦𝑢} → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢}))
23	19, 22	imbi12d 334	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = {𝑢 ∣ 𝑤𝑦𝑢} → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ({𝑢 ∣ 𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢})))
24	23	rspcv 3305	. . . . . . . . . . . . . 14 ⊢ ({𝑢 ∣ 𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦 → (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ({𝑢 ∣ 𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢})))
25	18, 24	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ({𝑢 ∣ 𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢}))
26	13, 25	syl5bi 232	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑤 ∈ dom 𝑦 → (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢}))
27	26	imp 445	. . . . . . . . . . 11 ⊢ ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢})
28		fvex 6201	. . . . . . . . . . . 12 ⊢ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ V
29		breq2 4657	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) → (𝑤𝑦𝑧 ↔ 𝑤𝑦(𝑓‘{𝑢 ∣ 𝑤𝑦𝑢})))
30		breq2 4657	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → (𝑤𝑦𝑢 ↔ 𝑤𝑦𝑧))
31	30	cbvabv 2747	. . . . . . . . . . . 12 ⊢ {𝑢 ∣ 𝑤𝑦𝑢} = {𝑧 ∣ 𝑤𝑦𝑧}
32	28, 29, 31	elab2 3354	. . . . . . . . . . 11 ⊢ ((𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢} ↔ 𝑤𝑦(𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))
33	27, 32	sylib 208	. . . . . . . . . 10 ⊢ ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → 𝑤𝑦(𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))
34		breq2 4657	. . . . . . . . . 10 ⊢ (ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) → (𝑤𝑦ℎ ↔ 𝑤𝑦(𝑓‘{𝑢 ∣ 𝑤𝑦𝑢})))
35	33, 34	syl5ibrcom 237	. . . . . . . . 9 ⊢ ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → (ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) → 𝑤𝑦ℎ))
36	35	expimpd 629	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑤 ∈ dom 𝑦 ∧ ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢})) → 𝑤𝑦ℎ))
37	36	ssopab2dv 5004	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → {⟨𝑤, ℎ⟩ ∣ (𝑤 ∈ dom 𝑦 ∧ ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))} ⊆ {⟨𝑤, ℎ⟩ ∣ 𝑤𝑦ℎ})
38		opabss 4714	. . . . . . 7 ⊢ {⟨𝑤, ℎ⟩ ∣ 𝑤𝑦ℎ} ⊆ 𝑦
39	37, 38	syl6ss 3615	. . . . . 6 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → {⟨𝑤, ℎ⟩ ∣ (𝑤 ∈ dom 𝑦 ∧ ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))} ⊆ 𝑦)
40	9, 39	syl5eqss 3649	. . . . 5 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → 𝐹 ⊆ 𝑦)
41	28, 7	fnmpti 6022	. . . . 5 ⊢ 𝐹 Fn dom 𝑦
42	1	ssex 4802	. . . . . . 7 ⊢ (𝐹 ⊆ 𝑦 → 𝐹 ∈ V)
43	42	adantr 481	. . . . . 6 ⊢ ((𝐹 ⊆ 𝑦 ∧ 𝐹 Fn dom 𝑦) → 𝐹 ∈ V)
44		sseq1 3626	. . . . . . . 8 ⊢ (𝑔 = 𝐹 → (𝑔 ⊆ 𝑦 ↔ 𝐹 ⊆ 𝑦))
45		fneq1 5979	. . . . . . . 8 ⊢ (𝑔 = 𝐹 → (𝑔 Fn dom 𝑦 ↔ 𝐹 Fn dom 𝑦))
46	44, 45	anbi12d 747	. . . . . . 7 ⊢ (𝑔 = 𝐹 → ((𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦) ↔ (𝐹 ⊆ 𝑦 ∧ 𝐹 Fn dom 𝑦)))
47	46	spcegv 3294	. . . . . 6 ⊢ (𝐹 ∈ V → ((𝐹 ⊆ 𝑦 ∧ 𝐹 Fn dom 𝑦) → ∃𝑔(𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦)))
48	43, 47	mpcom 38	. . . . 5 ⊢ ((𝐹 ⊆ 𝑦 ∧ 𝐹 Fn dom 𝑦) → ∃𝑔(𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦))
49	40, 41, 48	sylancl 694	. . . 4 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑔(𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦))
50	49	exlimiv 1858	. . 3 ⊢ (∃𝑓∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑔(𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦))
51	6, 50	syl 17	. 2 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑔(𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦))
52		sseq1 3626	. . . 4 ⊢ (𝑔 = 𝑓 → (𝑔 ⊆ 𝑦 ↔ 𝑓 ⊆ 𝑦))
53		fneq1 5979	. . . 4 ⊢ (𝑔 = 𝑓 → (𝑔 Fn dom 𝑦 ↔ 𝑓 Fn dom 𝑦))
54	52, 53	anbi12d 747	. . 3 ⊢ (𝑔 = 𝑓 → ((𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦) ↔ (𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦)))
55	54	cbvexv 2275	. 2 ⊢ (∃𝑔(𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦) ↔ ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
56	51, 55	sylib 208	1 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158   class class class wbr 4653  {copab 4712   ↦ cmpt 4729  dom cdm 5114  ran crn 5115   Fn wfn 5883  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  dfac3  8944