Theorem aceq3lem 10070

Description: Lemma for dfac3 10071. (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 3457	. . . . . 6 ⊢ 𝑦 ∈ V
2	1	rnex 7886	. . . . 5 ⊢ ran 𝑦 ∈ V
3	2	pwex 5334	. . . 4 ⊢ 𝒫 ran 𝑦 ∈ V
4		raleq 3316	. . . . 5 ⊢ (𝑥 = 𝒫 ran 𝑦 → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
5	4	exbidv 1940	. . . 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 3563	. . 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 5179	. . . . . . 7 ⊢ (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢})) = {⟨𝑤, ℎ⟩ ∣ (𝑤 ∈ dom 𝑦 ∧ ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))}
9	7, 8	eqtri 2784	. . . . . 6 ⊢ 𝐹 = {⟨𝑤, ℎ⟩ ∣ (𝑤 ∈ dom 𝑦 ∧ ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))}
10		vex 3457	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
11	10	eldm 5872	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ dom 𝑦 ↔ ∃𝑢 𝑤𝑦𝑢)
12		abn0 4335	. . . . . . . . . . . . . 14 ⊢ ({𝑢 ∣ 𝑤𝑦𝑢} ≠ ∅ ↔ ∃𝑢 𝑤𝑦𝑢)
13	11, 12	bitr4i 280	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ dom 𝑦 ↔ {𝑢 ∣ 𝑤𝑦𝑢} ≠ ∅)
14		vex 3457	. . . . . . . . . . . . . . . . 17 ⊢ 𝑢 ∈ V
15	10, 14	brelrn 5914	. . . . . . . . . . . . . . . 16 ⊢ (𝑤𝑦𝑢 → 𝑢 ∈ ran 𝑦)
16	15	abssi 4019	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ 𝑤𝑦𝑢} ⊆ ran 𝑦
17	2, 16	elpwi2 5288	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∣ 𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦
18		neeq1 3018	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = {𝑢 ∣ 𝑤𝑦𝑢} → (𝑧 ≠ ∅ ↔ {𝑢 ∣ 𝑤𝑦𝑢} ≠ ∅))
19		fveq2 6862	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = {𝑢 ∣ 𝑤𝑦𝑢} → (𝑓‘𝑧) = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))
20		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = {𝑢 ∣ 𝑤𝑦𝑢} → 𝑧 = {𝑢 ∣ 𝑤𝑦𝑢})
21	19, 20	eleq12d 2855	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = {𝑢 ∣ 𝑤𝑦𝑢} → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢}))
22	18, 21	imbi12d 346	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = {𝑢 ∣ 𝑤𝑦𝑢} → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ({𝑢 ∣ 𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢})))
23	22	rspcv 3576	. . . . . . . . . . . . . 14 ⊢ ({𝑢 ∣ 𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦 → (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ({𝑢 ∣ 𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢})))
24	17, 23	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ({𝑢 ∣ 𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢}))
25	13, 24	biimtrid 244	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑤 ∈ dom 𝑦 → (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢}))
26	25	imp 410	. . . . . . . . . . 11 ⊢ ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢})
27		fvex 6875	. . . . . . . . . . . 12 ⊢ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ V
28		breq2 5101	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) → (𝑤𝑦𝑧 ↔ 𝑤𝑦(𝑓‘{𝑢 ∣ 𝑤𝑦𝑢})))
29		breq2 5101	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → (𝑤𝑦𝑢 ↔ 𝑤𝑦𝑧))
30	29	cbvabv 2831	. . . . . . . . . . . 12 ⊢ {𝑢 ∣ 𝑤𝑦𝑢} = {𝑧 ∣ 𝑤𝑦𝑧}
31	27, 28, 30	elab2 3640	. . . . . . . . . . 11 ⊢ ((𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) ∈ {𝑢 ∣ 𝑤𝑦𝑢} ↔ 𝑤𝑦(𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))
32	26, 31	sylib 220	. . . . . . . . . 10 ⊢ ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → 𝑤𝑦(𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))
33		breq2 5101	. . . . . . . . . 10 ⊢ (ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) → (𝑤𝑦ℎ ↔ 𝑤𝑦(𝑓‘{𝑢 ∣ 𝑤𝑦𝑢})))
34	32, 33	syl5ibrcom 249	. . . . . . . . 9 ⊢ ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → (ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}) → 𝑤𝑦ℎ))
35	34	expimpd 457	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝑤 ∈ dom 𝑦 ∧ ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢})) → 𝑤𝑦ℎ))
36	35	ssopab2dv 5518	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → {⟨𝑤, ℎ⟩ ∣ (𝑤 ∈ dom 𝑦 ∧ ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))} ⊆ {⟨𝑤, ℎ⟩ ∣ 𝑤𝑦ℎ})
37		opabss 5161	. . . . . . 7 ⊢ {⟨𝑤, ℎ⟩ ∣ 𝑤𝑦ℎ} ⊆ 𝑦
38	36, 37	sstrdi 3946	. . . . . 6 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → {⟨𝑤, ℎ⟩ ∣ (𝑤 ∈ dom 𝑦 ∧ ℎ = (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))} ⊆ 𝑦)
39	9, 38	eqsstrid 3972	. . . . 5 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → 𝐹 ⊆ 𝑦)
40	27, 7	fnmpti 6659	. . . . 5 ⊢ 𝐹 Fn dom 𝑦
41	1	ssex 5274	. . . . . . 7 ⊢ (𝐹 ⊆ 𝑦 → 𝐹 ∈ V)
42	41	adantr 484	. . . . . 6 ⊢ ((𝐹 ⊆ 𝑦 ∧ 𝐹 Fn dom 𝑦) → 𝐹 ∈ V)
43		sseq1 3959	. . . . . . . 8 ⊢ (𝑔 = 𝐹 → (𝑔 ⊆ 𝑦 ↔ 𝐹 ⊆ 𝑦))
44		fneq1 6607	. . . . . . . 8 ⊢ (𝑔 = 𝐹 → (𝑔 Fn dom 𝑦 ↔ 𝐹 Fn dom 𝑦))
45	43, 44	anbi12d 641	. . . . . . 7 ⊢ (𝑔 = 𝐹 → ((𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦) ↔ (𝐹 ⊆ 𝑦 ∧ 𝐹 Fn dom 𝑦)))
46	45	spcegv 3555	. . . . . 6 ⊢ (𝐹 ∈ V → ((𝐹 ⊆ 𝑦 ∧ 𝐹 Fn dom 𝑦) → ∃𝑔(𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦)))
47	42, 46	mpcom 38	. . . . 5 ⊢ ((𝐹 ⊆ 𝑦 ∧ 𝐹 Fn dom 𝑦) → ∃𝑔(𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦))
48	39, 40, 47	sylancl 595	. . . 4 ⊢ (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑔(𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦))
49	48	exlimiv 1949	. . 3 ⊢ (∃𝑓∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑔(𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦))
50	6, 49	syl 17	. 2 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑔(𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦))
51		sseq1 3959	. . . 4 ⊢ (𝑔 = 𝑓 → (𝑔 ⊆ 𝑦 ↔ 𝑓 ⊆ 𝑦))
52		fneq1 6607	. . . 4 ⊢ (𝑔 = 𝑓 → (𝑔 Fn dom 𝑦 ↔ 𝑓 Fn dom 𝑦))
53	51, 52	anbi12d 641	. . 3 ⊢ (𝑔 = 𝑓 → ((𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦) ↔ (𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦)))
54	53	cbvexvw 2056	. 2 ⊢ (∃𝑔(𝑔 ⊆ 𝑦 ∧ 𝑔 Fn dom 𝑦) ↔ ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
55	50, 54	sylib 220	1 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739   ≠ wne 2956  ∀wral 3075  Vcvv 3453   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4552   class class class wbr 5097  {copab 5159   ↦ cmpt 5178  dom cdm 5643  ran crn 5644   Fn wfn 6511  ‘cfv 6516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-iota 6472  df-fun 6518  df-fn 6519  df-fv 6524

This theorem is referenced by:  dfac3  10071