fin23lem32 - Metamath Proof Explorer

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Theorem fin23lem32 10341

Description: Lemma for fin23 10386. Wrap the previous construction into a function to hide the hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)

**Hypotheses**
Ref	Expression
fin23lem.a	⊢ 𝑈 = seq_ω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
fin23lem17.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑_m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
fin23lem.b	⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)}
fin23lem.c	⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤))
fin23lem.d	⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
fin23lem.e	⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))

**Assertion**
Ref	Expression
fin23lem32	⊢ (𝐺 ∈ 𝐹 → ∃𝑓∀𝑏((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) → ((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏)))

Distinct variable groups: 𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧 𝑎,𝑏,𝑖,𝑢,𝑡 𝐹,𝑎,𝑡 𝑤,𝑎,𝑥,𝑧,𝑃,𝑏 𝑣,𝑎,𝑅,𝑏,𝑖,𝑢 𝑈,𝑎,𝑏,𝑖,𝑢,𝑣,𝑧 𝑓,𝑎,𝑍,𝑏 𝑔,𝑎,𝐺,𝑏,𝑡,𝑓,𝑥 Allowed substitution hints: 𝑃(𝑣,𝑢,𝑡,𝑓,𝑔,𝑖) 𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑓,𝑔,𝑖,𝑎,𝑏) 𝑅(𝑥,𝑧,𝑤,𝑡,𝑓,𝑔) 𝑈(𝑥,𝑤,𝑡,𝑓,𝑔) 𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑓,𝑔,𝑖,𝑏) 𝐺(𝑧,𝑤,𝑣,𝑢,𝑖) 𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

**Proof of Theorem fin23lem32**
Step	Hyp	Ref	Expression
1		fin23lem.a	. . . . . . . 8 ⊢ 𝑈 = seq_ω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
2		fin23lem17.f	. . . . . . . 8 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑_m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
3		fin23lem.b	. . . . . . . 8 ⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)}
4		fin23lem.c	. . . . . . . 8 ⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤))
5		fin23lem.d	. . . . . . . 8 ⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
6		fin23lem.e	. . . . . . . 8 ⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
7	1, 2, 3, 4, 5, 6	fin23lem28 10337	. . . . . . 7 ⊢ (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)
8	7	ad2antrl 725	. . . . . 6 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → 𝑍:ω–1-1→V)
9		simprl 768	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → 𝑡:ω–1-1→V)
10		simpl 482	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → 𝐺 ∈ 𝐹)
11		simprr 770	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ∪ ran 𝑡 ⊆ 𝐺)
12	1, 2, 3, 4, 5, 6	fin23lem31 10340	. . . . . . 7 ⊢ ((𝑡:ω–1-1→V ∧ 𝐺 ∈ 𝐹 ∧ ∪ ran 𝑡 ⊆ 𝐺) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)
13	9, 10, 11, 12	syl3anc 1368	. . . . . 6 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)
14		f1fn 6782	. . . . . . . . . . . 12 ⊢ (𝑡:ω–1-1→V → 𝑡 Fn ω)
15		dffn3 6724	. . . . . . . . . . . 12 ⊢ (𝑡 Fn ω ↔ 𝑡:ω⟶ran 𝑡)
16	14, 15	sylib 217	. . . . . . . . . . 11 ⊢ (𝑡:ω–1-1→V → 𝑡:ω⟶ran 𝑡)
17	16	ad2antrl 725	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → 𝑡:ω⟶ran 𝑡)
18		sspwuni 5096	. . . . . . . . . . . 12 ⊢ (ran 𝑡 ⊆ 𝒫 𝐺 ↔ ∪ ran 𝑡 ⊆ 𝐺)
19	18	biimpri 227	. . . . . . . . . . 11 ⊢ (∪ ran 𝑡 ⊆ 𝐺 → ran 𝑡 ⊆ 𝒫 𝐺)
20	19	ad2antll 726	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ran 𝑡 ⊆ 𝒫 𝐺)
21	17, 20	fssd 6729	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → 𝑡:ω⟶𝒫 𝐺)
22		pwexg 5369	. . . . . . . . . . 11 ⊢ (𝐺 ∈ 𝐹 → 𝒫 𝐺 ∈ V)
23	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → 𝒫 𝐺 ∈ V)
24		vex 3472	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
25		f1f 6781	. . . . . . . . . . . 12 ⊢ (𝑡:ω–1-1→V → 𝑡:ω⟶V)
26		dmfex 7895	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ V ∧ 𝑡:ω⟶V) → ω ∈ V)
27	24, 25, 26	sylancr 586	. . . . . . . . . . 11 ⊢ (𝑡:ω–1-1→V → ω ∈ V)
28	27	ad2antrl 725	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ω ∈ V)
29	23, 28	elmapd 8836	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → (𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↔ 𝑡:ω⟶𝒫 𝐺))
30	21, 29	mpbird 257	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → 𝑡 ∈ (𝒫 𝐺 ↑_m ω))
31		f1f 6781	. . . . . . . . . 10 ⊢ (𝑍:ω–1-1→V → 𝑍:ω⟶V)
32	8, 31	syl 17	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → 𝑍:ω⟶V)
33	32, 28	fexd 7224	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → 𝑍 ∈ V)
34		eqid 2726	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍) = (𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)
35	34	fvmpt2 7003	. . . . . . . 8 ⊢ ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ∧ 𝑍 ∈ V) → ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) = 𝑍)
36	30, 33, 35	syl2anc 583	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) = 𝑍)
37		f1eq1 6776	. . . . . . . 8 ⊢ (((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) = 𝑍 → (((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ↔ 𝑍:ω–1-1→V))
38		rneq 5929	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) = 𝑍 → ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) = ran 𝑍)
39	38	unieqd 4915	. . . . . . . . 9 ⊢ (((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) = 𝑍 → ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) = ∪ ran 𝑍)
40	39	psseq1d 4087	. . . . . . . 8 ⊢ (((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) = 𝑍 → (∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) ⊊ ∪ ran 𝑡 ↔ ∪ ran 𝑍 ⊊ ∪ ran 𝑡))
41	37, 40	anbi12d 630	. . . . . . 7 ⊢ (((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) = 𝑍 → ((((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) ⊊ ∪ ran 𝑡) ↔ (𝑍:ω–1-1→V ∧ ∪ ran 𝑍 ⊊ ∪ ran 𝑡)))
42	36, 41	syl 17	. . . . . 6 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ((((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) ⊊ ∪ ran 𝑡) ↔ (𝑍:ω–1-1→V ∧ ∪ ran 𝑍 ⊊ ∪ ran 𝑡)))
43	8, 13, 42	mpbir2and 710	. . . . 5 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → (((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) ⊊ ∪ ran 𝑡))
44	43	ex 412	. . . 4 ⊢ (𝐺 ∈ 𝐹 → ((𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺) → (((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) ⊊ ∪ ran 𝑡)))
45	44	alrimiv 1922	. . 3 ⊢ (𝐺 ∈ 𝐹 → ∀𝑡((𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺) → (((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) ⊊ ∪ ran 𝑡)))
46		ovex 7438	. . . . 5 ⊢ (𝒫 𝐺 ↑_m ω) ∈ V
47	46	mptex 7220	. . . 4 ⊢ (𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍) ∈ V
48		nfmpt1 5249	. . . . . 6 ⊢ Ⅎ𝑡(𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)
49	48	nfeq2 2914	. . . . 5 ⊢ Ⅎ𝑡 𝑓 = (𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)
50		fveq1 6884	. . . . . . . 8 ⊢ (𝑓 = (𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍) → (𝑓‘𝑡) = ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡))
51		f1eq1 6776	. . . . . . . 8 ⊢ ((𝑓‘𝑡) = ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) → ((𝑓‘𝑡):ω–1-1→V ↔ ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡):ω–1-1→V))
52	50, 51	syl 17	. . . . . . 7 ⊢ (𝑓 = (𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍) → ((𝑓‘𝑡):ω–1-1→V ↔ ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡):ω–1-1→V))
53	50	rneqd 5931	. . . . . . . . 9 ⊢ (𝑓 = (𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍) → ran (𝑓‘𝑡) = ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡))
54	53	unieqd 4915	. . . . . . . 8 ⊢ (𝑓 = (𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍) → ∪ ran (𝑓‘𝑡) = ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡))
55	54	psseq1d 4087	. . . . . . 7 ⊢ (𝑓 = (𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍) → (∪ ran (𝑓‘𝑡) ⊊ ∪ ran 𝑡 ↔ ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) ⊊ ∪ ran 𝑡))
56	52, 55	anbi12d 630	. . . . . 6 ⊢ (𝑓 = (𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍) → (((𝑓‘𝑡):ω–1-1→V ∧ ∪ ran (𝑓‘𝑡) ⊊ ∪ ran 𝑡) ↔ (((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) ⊊ ∪ ran 𝑡)))
57	56	imbi2d 340	. . . . 5 ⊢ (𝑓 = (𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍) → (((𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺) → ((𝑓‘𝑡):ω–1-1→V ∧ ∪ ran (𝑓‘𝑡) ⊊ ∪ ran 𝑡)) ↔ ((𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺) → (((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) ⊊ ∪ ran 𝑡))))
58	49, 57	albid 2207	. . . 4 ⊢ (𝑓 = (𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍) → (∀𝑡((𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺) → ((𝑓‘𝑡):ω–1-1→V ∧ ∪ ran (𝑓‘𝑡) ⊊ ∪ ran 𝑡)) ↔ ∀𝑡((𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺) → (((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) ⊊ ∪ ran 𝑡))))
59	47, 58	spcev 3590	. . 3 ⊢ (∀𝑡((𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺) → (((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑_m ω) ↦ 𝑍)‘𝑡) ⊊ ∪ ran 𝑡)) → ∃𝑓∀𝑡((𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺) → ((𝑓‘𝑡):ω–1-1→V ∧ ∪ ran (𝑓‘𝑡) ⊊ ∪ ran 𝑡)))
60	45, 59	syl 17	. 2 ⊢ (𝐺 ∈ 𝐹 → ∃𝑓∀𝑡((𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺) → ((𝑓‘𝑡):ω–1-1→V ∧ ∪ ran (𝑓‘𝑡) ⊊ ∪ ran 𝑡)))
61		f1eq1 6776	. . . . . 6 ⊢ (𝑏 = 𝑡 → (𝑏:ω–1-1→V ↔ 𝑡:ω–1-1→V))
62		rneq 5929	. . . . . . . 8 ⊢ (𝑏 = 𝑡 → ran 𝑏 = ran 𝑡)
63	62	unieqd 4915	. . . . . . 7 ⊢ (𝑏 = 𝑡 → ∪ ran 𝑏 = ∪ ran 𝑡)
64	63	sseq1d 4008	. . . . . 6 ⊢ (𝑏 = 𝑡 → (∪ ran 𝑏 ⊆ 𝐺 ↔ ∪ ran 𝑡 ⊆ 𝐺))
65	61, 64	anbi12d 630	. . . . 5 ⊢ (𝑏 = 𝑡 → ((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) ↔ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)))
66		fveq2 6885	. . . . . . 7 ⊢ (𝑏 = 𝑡 → (𝑓‘𝑏) = (𝑓‘𝑡))
67		f1eq1 6776	. . . . . . 7 ⊢ ((𝑓‘𝑏) = (𝑓‘𝑡) → ((𝑓‘𝑏):ω–1-1→V ↔ (𝑓‘𝑡):ω–1-1→V))
68	66, 67	syl 17	. . . . . 6 ⊢ (𝑏 = 𝑡 → ((𝑓‘𝑏):ω–1-1→V ↔ (𝑓‘𝑡):ω–1-1→V))
69	66	rneqd 5931	. . . . . . . 8 ⊢ (𝑏 = 𝑡 → ran (𝑓‘𝑏) = ran (𝑓‘𝑡))
70	69	unieqd 4915	. . . . . . 7 ⊢ (𝑏 = 𝑡 → ∪ ran (𝑓‘𝑏) = ∪ ran (𝑓‘𝑡))
71	70, 63	psseq12d 4089	. . . . . 6 ⊢ (𝑏 = 𝑡 → (∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏 ↔ ∪ ran (𝑓‘𝑡) ⊊ ∪ ran 𝑡))
72	68, 71	anbi12d 630	. . . . 5 ⊢ (𝑏 = 𝑡 → (((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏) ↔ ((𝑓‘𝑡):ω–1-1→V ∧ ∪ ran (𝑓‘𝑡) ⊊ ∪ ran 𝑡)))
73	65, 72	imbi12d 344	. . . 4 ⊢ (𝑏 = 𝑡 → (((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) → ((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏)) ↔ ((𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺) → ((𝑓‘𝑡):ω–1-1→V ∧ ∪ ran (𝑓‘𝑡) ⊊ ∪ ran 𝑡))))
74	73	cbvalvw 2031	. . 3 ⊢ (∀𝑏((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) → ((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏)) ↔ ∀𝑡((𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺) → ((𝑓‘𝑡):ω–1-1→V ∧ ∪ ran (𝑓‘𝑡) ⊊ ∪ ran 𝑡)))
75	74	exbii 1842	. 2 ⊢ (∃𝑓∀𝑏((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) → ((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏)) ↔ ∃𝑓∀𝑡((𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺) → ((𝑓‘𝑡):ω–1-1→V ∧ ∪ ran (𝑓‘𝑡) ⊊ ∪ ran 𝑡)))
76	60, 75	sylibr 233	1 ⊢ (𝐺 ∈ 𝐹 → ∃𝑓∀𝑏((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) → ((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2703 ∀wral 3055 {crab 3426 Vcvv 3468 ∖ cdif 3940 ∩ cin 3942 ⊆ wss 3943 ⊊ wpss 3944 ∅c0 4317 ifcif 4523 𝒫 cpw 4597 ∪ cuni 4902 ∩ cint 4943 class class class wbr 5141 ↦ cmpt 5224 ran crn 5670 ∘ ccom 5673 suc csuc 6360 Fn wfn 6532 ⟶wf 6533 –1-1→wf1 6534 ‘cfv 6537 ℩crio 7360 (class class class)co 7405 ∈ cmpo 7407 ωcom 7852 seq_ωcseqom 8448 ↑_m cmap 8822 ≈ cen 8938 Fincfn 8941

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-seqom 8449 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936

This theorem is referenced by: fin23lem33 10342

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