Theorem fin23lem32 10297

Description: Lemma for fin23 10342. 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 = seqω((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 = seqω((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 10293	. . . . . . 7 ⊢ (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)
8	7	ad2antrl 728	. . . . . 6 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → 𝑍:ω–1-1→V)
9		simprl 770	. . . . . . 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 772	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ∪ ran 𝑡 ⊆ 𝐺)
12	1, 2, 3, 4, 5, 6	fin23lem31 10296	. . . . . . 7 ⊢ ((𝑡:ω–1-1→V ∧ 𝐺 ∈ 𝐹 ∧ ∪ ran 𝑡 ⊆ 𝐺) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)
13	9, 10, 11, 12	syl3anc 1373	. . . . . 6 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)
14		f1fn 6757	. . . . . . . . . . . 12 ⊢ (𝑡:ω–1-1→V → 𝑡 Fn ω)
15		dffn3 6700	. . . . . . . . . . . 12 ⊢ (𝑡 Fn ω ↔ 𝑡:ω⟶ran 𝑡)
16	14, 15	sylib 218	. . . . . . . . . . 11 ⊢ (𝑡:ω–1-1→V → 𝑡:ω⟶ran 𝑡)
17	16	ad2antrl 728	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → 𝑡:ω⟶ran 𝑡)
18		sspwuni 5064	. . . . . . . . . . . 12 ⊢ (ran 𝑡 ⊆ 𝒫 𝐺 ↔ ∪ ran 𝑡 ⊆ 𝐺)
19	18	biimpri 228	. . . . . . . . . . 11 ⊢ (∪ ran 𝑡 ⊆ 𝐺 → ran 𝑡 ⊆ 𝒫 𝐺)
20	19	ad2antll 729	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ran 𝑡 ⊆ 𝒫 𝐺)
21	17, 20	fssd 6705	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → 𝑡:ω⟶𝒫 𝐺)
22		pwexg 5333	. . . . . . . . . . 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 3451	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
25		f1f 6756	. . . . . . . . . . . 12 ⊢ (𝑡:ω–1-1→V → 𝑡:ω⟶V)
26		dmfex 7881	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ V ∧ 𝑡:ω⟶V) → ω ∈ V)
27	24, 25, 26	sylancr 587	. . . . . . . . . . 11 ⊢ (𝑡:ω–1-1→V → ω ∈ V)
28	27	ad2antrl 728	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ω ∈ V)
29	23, 28	elmapd 8813	. . . . . . . . 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 6756	. . . . . . . . . 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 7201	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → 𝑍 ∈ V)
34		eqid 2729	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍) = (𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍)
35	34	fvmpt2 6979	. . . . . . . 8 ⊢ ((𝑡 ∈ (𝒫 𝐺 ↑m ω) ∧ 𝑍 ∈ V) → ((𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍)‘𝑡) = 𝑍)
36	30, 33, 35	syl2anc 584	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐹 ∧ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ((𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍)‘𝑡) = 𝑍)
37		f1eq1 6751	. . . . . . . 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 5900	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍)‘𝑡) = 𝑍 → ran ((𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍)‘𝑡) = ran 𝑍)
39	38	unieqd 4884	. . . . . . . . 9 ⊢ (((𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍)‘𝑡) = 𝑍 → ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍)‘𝑡) = ∪ ran 𝑍)
40	39	psseq1d 4058	. . . . . . . 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 632	. . . . . . 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 713	. . . . 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 1927	. . 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 7420	. . . . 5 ⊢ (𝒫 𝐺 ↑m ω) ∈ V
47	46	mptex 7197	. . . 4 ⊢ (𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍) ∈ V
48		nfmpt1 5206	. . . . . 6 ⊢ Ⅎ𝑡(𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍)
49	48	nfeq2 2909	. . . . 5 ⊢ Ⅎ𝑡 𝑓 = (𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍)
50		fveq1 6857	. . . . . . . 8 ⊢ (𝑓 = (𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍) → (𝑓‘𝑡) = ((𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍)‘𝑡))
51		f1eq1 6751	. . . . . . . 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 5902	. . . . . . . . 9 ⊢ (𝑓 = (𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍) → ran (𝑓‘𝑡) = ran ((𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍)‘𝑡))
54	53	unieqd 4884	. . . . . . . 8 ⊢ (𝑓 = (𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍) → ∪ ran (𝑓‘𝑡) = ∪ ran ((𝑡 ∈ (𝒫 𝐺 ↑m ω) ↦ 𝑍)‘𝑡))
55	54	psseq1d 4058	. . . . . . 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 632	. . . . . 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 2223	. . . 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 3572	. . 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 6751	. . . . . 6 ⊢ (𝑏 = 𝑡 → (𝑏:ω–1-1→V ↔ 𝑡:ω–1-1→V))
62		rneq 5900	. . . . . . . 8 ⊢ (𝑏 = 𝑡 → ran 𝑏 = ran 𝑡)
63	62	unieqd 4884	. . . . . . 7 ⊢ (𝑏 = 𝑡 → ∪ ran 𝑏 = ∪ ran 𝑡)
64	63	sseq1d 3978	. . . . . 6 ⊢ (𝑏 = 𝑡 → (∪ ran 𝑏 ⊆ 𝐺 ↔ ∪ ran 𝑡 ⊆ 𝐺))
65	61, 64	anbi12d 632	. . . . 5 ⊢ (𝑏 = 𝑡 → ((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) ↔ (𝑡:ω–1-1→V ∧ ∪ ran 𝑡 ⊆ 𝐺)))
66		fveq2 6858	. . . . . . 7 ⊢ (𝑏 = 𝑡 → (𝑓‘𝑏) = (𝑓‘𝑡))
67		f1eq1 6751	. . . . . . 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 5902	. . . . . . . 8 ⊢ (𝑏 = 𝑡 → ran (𝑓‘𝑏) = ran (𝑓‘𝑡))
70	69	unieqd 4884	. . . . . . 7 ⊢ (𝑏 = 𝑡 → ∪ ran (𝑓‘𝑏) = ∪ ran (𝑓‘𝑡))
71	70, 63	psseq12d 4060	. . . . . 6 ⊢ (𝑏 = 𝑡 → (∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏 ↔ ∪ ran (𝑓‘𝑡) ⊊ ∪ ran 𝑡))
72	68, 71	anbi12d 632	. . . . 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 2036	. . 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 1848	. 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 234	1 ⊢ (𝐺 ∈ 𝐹 → ∃𝑓∀𝑏((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) → ((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∀wral 3044  {crab 3405  Vcvv 3447   ∖ cdif 3911   ∩ cin 3913   ⊆ wss 3914   ⊊ wpss 3915  ∅c0 4296  ifcif 4488  𝒫 cpw 4563  ∪ cuni 4871  ∩ cint 4910   class class class wbr 5107   ↦ cmpt 5188  ran crn 5639   ∘ ccom 5642  suc csuc 6334   Fn wfn 6506  ⟶wf 6507  –1-1→wf1 6508  ‘cfv 6511  ℩crio 7343  (class class class)co 7387   ∈ cmpo 7389  ωcom 7842  seq_ωcseqom 8415   ↑_m cmap 8799   ≈ cen 8915  Fincfn 8918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-seqom 8416  df-1o 8434  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-card 9892

This theorem is referenced by:  fin23lem33  10298