Theorem fin23lem28 9162

Description: Lemma for fin23 9211. The residual is also one-to-one. This preserves the induction invariant. (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
fin23lem28	⊢ (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)

Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧,𝑎   𝐹,𝑎,𝑡   𝑤,𝑎,𝑥,𝑧,𝑃   𝑣,𝑎,𝑅,𝑖,𝑢   𝑈,𝑎,𝑖,𝑢,𝑣,𝑧   𝑍,𝑎   𝑔,𝑎

Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑔,𝑖)   𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖,𝑎)   𝑅(𝑥,𝑧,𝑤,𝑡,𝑔)   𝑈(𝑥,𝑤,𝑡,𝑔)   𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑔,𝑖)   𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem28

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin23lem.e	. . 3 ⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
2		eqif 4126	. . 3 ⊢ (𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))))
3	1, 2	mpbi 220	. 2 ⊢ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)))
4		difss 3737	. . . . . . . . 9 ⊢ (ω ∖ 𝑃) ⊆ ω
5		ominf 8172	. . . . . . . . . 10 ⊢ ¬ ω ∈ Fin
6		fin23lem.b	. . . . . . . . . . . . . 14 ⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)}
7		ssrab2 3687	. . . . . . . . . . . . . 14 ⊢ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} ⊆ ω
8	6, 7	eqsstri 3635	. . . . . . . . . . . . 13 ⊢ 𝑃 ⊆ ω
9		undif 4049	. . . . . . . . . . . . 13 ⊢ (𝑃 ⊆ ω ↔ (𝑃 ∪ (ω ∖ 𝑃)) = ω)
10	8, 9	mpbi 220	. . . . . . . . . . . 12 ⊢ (𝑃 ∪ (ω ∖ 𝑃)) = ω
11		unfi 8227	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → (𝑃 ∪ (ω ∖ 𝑃)) ∈ Fin)
12	10, 11	syl5eqelr 2706	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → ω ∈ Fin)
13	12	ex 450	. . . . . . . . . 10 ⊢ (𝑃 ∈ Fin → ((ω ∖ 𝑃) ∈ Fin → ω ∈ Fin))
14	5, 13	mtoi 190	. . . . . . . . 9 ⊢ (𝑃 ∈ Fin → ¬ (ω ∖ 𝑃) ∈ Fin)
15		fin23lem.d	. . . . . . . . . 10 ⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
16	15	fin23lem22 9149	. . . . . . . . 9 ⊢ (((ω ∖ 𝑃) ⊆ ω ∧ ¬ (ω ∖ 𝑃) ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
17	4, 14, 16	sylancr 695	. . . . . . . 8 ⊢ (𝑃 ∈ Fin → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
18	17	adantl 482	. . . . . . 7 ⊢ ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
19		f1of1 6136	. . . . . . 7 ⊢ (𝑅:ω–1-1-onto→(ω ∖ 𝑃) → 𝑅:ω–1-1→(ω ∖ 𝑃))
20		f1ss 6106	. . . . . . . 8 ⊢ ((𝑅:ω–1-1→(ω ∖ 𝑃) ∧ (ω ∖ 𝑃) ⊆ ω) → 𝑅:ω–1-1→ω)
21	4, 20	mpan2 707	. . . . . . 7 ⊢ (𝑅:ω–1-1→(ω ∖ 𝑃) → 𝑅:ω–1-1→ω)
22	18, 19, 21	3syl 18	. . . . . 6 ⊢ ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → 𝑅:ω–1-1→ω)
23		f1co 6110	. . . . . 6 ⊢ ((𝑡:ω–1-1→V ∧ 𝑅:ω–1-1→ω) → (𝑡 ∘ 𝑅):ω–1-1→V)
24	22, 23	syldan 487	. . . . 5 ⊢ ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → (𝑡 ∘ 𝑅):ω–1-1→V)
25		f1eq1 6096	. . . . 5 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → (𝑍:ω–1-1→V ↔ (𝑡 ∘ 𝑅):ω–1-1→V))
26	24, 25	syl5ibrcom 237	. . . 4 ⊢ ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → (𝑍 = (𝑡 ∘ 𝑅) → 𝑍:ω–1-1→V))
27	26	impr 649	. . 3 ⊢ ((𝑡:ω–1-1→V ∧ (𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅))) → 𝑍:ω–1-1→V)
28		fvex 6201	. . . . . . . . . . 11 ⊢ (𝑡‘𝑧) ∈ V
29		difexg 4808	. . . . . . . . . . 11 ⊢ ((𝑡‘𝑧) ∈ V → ((𝑡‘𝑧) ∖ ∩ ran 𝑈) ∈ V)
30	28, 29	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑡‘𝑧) ∖ ∩ ran 𝑈) ∈ V
31	30	rgenw 2924	. . . . . . . . 9 ⊢ ∀𝑧 ∈ 𝑃 ((𝑡‘𝑧) ∖ ∩ ran 𝑈) ∈ V
32		eqid 2622	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
33	32	fmpt 6381	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑃 ((𝑡‘𝑧) ∖ ∩ ran 𝑈) ∈ V ↔ (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)):𝑃⟶V)
34	31, 33	mpbi 220	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)):𝑃⟶V
35	34	a1i 11	. . . . . . 7 ⊢ (𝑡:ω–1-1→V → (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)):𝑃⟶V)
36		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑎 → (𝑡‘𝑧) = (𝑡‘𝑎))
37	36	difeq1d 3727	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑎 → ((𝑡‘𝑧) ∖ ∩ ran 𝑈) = ((𝑡‘𝑎) ∖ ∩ ran 𝑈))
38		fvex 6201	. . . . . . . . . . . . 13 ⊢ (𝑡‘𝑎) ∈ V
39		difexg 4808	. . . . . . . . . . . . 13 ⊢ ((𝑡‘𝑎) ∈ V → ((𝑡‘𝑎) ∖ ∩ ran 𝑈) ∈ V)
40	38, 39	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑡‘𝑎) ∖ ∩ ran 𝑈) ∈ V
41	37, 32, 40	fvmpt 6282	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝑃 → ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))‘𝑎) = ((𝑡‘𝑎) ∖ ∩ ran 𝑈))
42	41	ad2antrl 764	. . . . . . . . . 10 ⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))‘𝑎) = ((𝑡‘𝑎) ∖ ∩ ran 𝑈))
43		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑏 → (𝑡‘𝑧) = (𝑡‘𝑏))
44	43	difeq1d 3727	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑏 → ((𝑡‘𝑧) ∖ ∩ ran 𝑈) = ((𝑡‘𝑏) ∖ ∩ ran 𝑈))
45		fvex 6201	. . . . . . . . . . . . 13 ⊢ (𝑡‘𝑏) ∈ V
46		difexg 4808	. . . . . . . . . . . . 13 ⊢ ((𝑡‘𝑏) ∈ V → ((𝑡‘𝑏) ∖ ∩ ran 𝑈) ∈ V)
47	45, 46	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑡‘𝑏) ∖ ∩ ran 𝑈) ∈ V
48	44, 32, 47	fvmpt 6282	. . . . . . . . . . 11 ⊢ (𝑏 ∈ 𝑃 → ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))‘𝑏) = ((𝑡‘𝑏) ∖ ∩ ran 𝑈))
49	48	ad2antll 765	. . . . . . . . . 10 ⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))‘𝑏) = ((𝑡‘𝑏) ∖ ∩ ran 𝑈))
50	42, 49	eqeq12d 2637	. . . . . . . . 9 ⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))‘𝑎) = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))‘𝑏) ↔ ((𝑡‘𝑎) ∖ ∩ ran 𝑈) = ((𝑡‘𝑏) ∖ ∩ ran 𝑈)))
51		uneq2 3761	. . . . . . . . . . 11 ⊢ (((𝑡‘𝑎) ∖ ∩ ran 𝑈) = ((𝑡‘𝑏) ∖ ∩ ran 𝑈) → (∩ ran 𝑈 ∪ ((𝑡‘𝑎) ∖ ∩ ran 𝑈)) = (∩ ran 𝑈 ∪ ((𝑡‘𝑏) ∖ ∩ ran 𝑈)))
52		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑎 → (𝑡‘𝑣) = (𝑡‘𝑎))
53	52	sseq2d 3633	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑎 → (∩ ran 𝑈 ⊆ (𝑡‘𝑣) ↔ ∩ ran 𝑈 ⊆ (𝑡‘𝑎)))
54	53, 6	elrab2 3366	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝑃 ↔ (𝑎 ∈ ω ∧ ∩ ran 𝑈 ⊆ (𝑡‘𝑎)))
55	54	simprbi 480	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝑃 → ∩ ran 𝑈 ⊆ (𝑡‘𝑎))
56	55	ad2antrl 764	. . . . . . . . . . . . 13 ⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ∩ ran 𝑈 ⊆ (𝑡‘𝑎))
57		undif 4049	. . . . . . . . . . . . 13 ⊢ (∩ ran 𝑈 ⊆ (𝑡‘𝑎) ↔ (∩ ran 𝑈 ∪ ((𝑡‘𝑎) ∖ ∩ ran 𝑈)) = (𝑡‘𝑎))
58	56, 57	sylib 208	. . . . . . . . . . . 12 ⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (∩ ran 𝑈 ∪ ((𝑡‘𝑎) ∖ ∩ ran 𝑈)) = (𝑡‘𝑎))
59		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑏 → (𝑡‘𝑣) = (𝑡‘𝑏))
60	59	sseq2d 3633	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑏 → (∩ ran 𝑈 ⊆ (𝑡‘𝑣) ↔ ∩ ran 𝑈 ⊆ (𝑡‘𝑏)))
61	60, 6	elrab2 3366	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ 𝑃 ↔ (𝑏 ∈ ω ∧ ∩ ran 𝑈 ⊆ (𝑡‘𝑏)))
62	61	simprbi 480	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝑃 → ∩ ran 𝑈 ⊆ (𝑡‘𝑏))
63	62	ad2antll 765	. . . . . . . . . . . . 13 ⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ∩ ran 𝑈 ⊆ (𝑡‘𝑏))
64		undif 4049	. . . . . . . . . . . . 13 ⊢ (∩ ran 𝑈 ⊆ (𝑡‘𝑏) ↔ (∩ ran 𝑈 ∪ ((𝑡‘𝑏) ∖ ∩ ran 𝑈)) = (𝑡‘𝑏))
65	63, 64	sylib 208	. . . . . . . . . . . 12 ⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (∩ ran 𝑈 ∪ ((𝑡‘𝑏) ∖ ∩ ran 𝑈)) = (𝑡‘𝑏))
66	58, 65	eqeq12d 2637	. . . . . . . . . . 11 ⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((∩ ran 𝑈 ∪ ((𝑡‘𝑎) ∖ ∩ ran 𝑈)) = (∩ ran 𝑈 ∪ ((𝑡‘𝑏) ∖ ∩ ran 𝑈)) ↔ (𝑡‘𝑎) = (𝑡‘𝑏)))
67	51, 66	syl5ib 234	. . . . . . . . . 10 ⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (((𝑡‘𝑎) ∖ ∩ ran 𝑈) = ((𝑡‘𝑏) ∖ ∩ ran 𝑈) → (𝑡‘𝑎) = (𝑡‘𝑏)))
68	8	sseli 3599	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑃 → 𝑎 ∈ ω)
69	8	sseli 3599	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ 𝑃 → 𝑏 ∈ ω)
70	68, 69	anim12i 590	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) → (𝑎 ∈ ω ∧ 𝑏 ∈ ω))
71		f1fveq 6519	. . . . . . . . . . 11 ⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ ω ∧ 𝑏 ∈ ω)) → ((𝑡‘𝑎) = (𝑡‘𝑏) ↔ 𝑎 = 𝑏))
72	70, 71	sylan2 491	. . . . . . . . . 10 ⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝑡‘𝑎) = (𝑡‘𝑏) ↔ 𝑎 = 𝑏))
73	67, 72	sylibd 229	. . . . . . . . 9 ⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (((𝑡‘𝑎) ∖ ∩ ran 𝑈) = ((𝑡‘𝑏) ∖ ∩ ran 𝑈) → 𝑎 = 𝑏))
74	50, 73	sylbid 230	. . . . . . . 8 ⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))‘𝑎) = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))‘𝑏) → 𝑎 = 𝑏))
75	74	ralrimivva 2971	. . . . . . 7 ⊢ (𝑡:ω–1-1→V → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))‘𝑎) = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))‘𝑏) → 𝑎 = 𝑏))
76		dff13 6512	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)):𝑃–1-1→V ↔ ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)):𝑃⟶V ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))‘𝑎) = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))‘𝑏) → 𝑎 = 𝑏)))
77	35, 75, 76	sylanbrc 698	. . . . . 6 ⊢ (𝑡:ω–1-1→V → (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)):𝑃–1-1→V)
78		fin23lem.c	. . . . . . . . 9 ⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤))
79	78	fin23lem22 9149	. . . . . . . 8 ⊢ ((𝑃 ⊆ ω ∧ ¬ 𝑃 ∈ Fin) → 𝑄:ω–1-1-onto→𝑃)
80		f1of1 6136	. . . . . . . 8 ⊢ (𝑄:ω–1-1-onto→𝑃 → 𝑄:ω–1-1→𝑃)
81	79, 80	syl 17	. . . . . . 7 ⊢ ((𝑃 ⊆ ω ∧ ¬ 𝑃 ∈ Fin) → 𝑄:ω–1-1→𝑃)
82	8, 81	mpan 706	. . . . . 6 ⊢ (¬ 𝑃 ∈ Fin → 𝑄:ω–1-1→𝑃)
83		f1co 6110	. . . . . 6 ⊢ (((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)):𝑃–1-1→V ∧ 𝑄:ω–1-1→𝑃) → ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄):ω–1-1→V)
84	77, 82, 83	syl2an 494	. . . . 5 ⊢ ((𝑡:ω–1-1→V ∧ ¬ 𝑃 ∈ Fin) → ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄):ω–1-1→V)
85		f1eq1 6096	. . . . 5 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → (𝑍:ω–1-1→V ↔ ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄):ω–1-1→V))
86	84, 85	syl5ibrcom 237	. . . 4 ⊢ ((𝑡:ω–1-1→V ∧ ¬ 𝑃 ∈ Fin) → (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → 𝑍:ω–1-1→V))
87	86	impr 649	. . 3 ⊢ ((𝑡:ω–1-1→V ∧ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))) → 𝑍:ω–1-1→V)
88	27, 87	jaodan 826	. 2 ⊢ ((𝑡:ω–1-1→V ∧ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)))) → 𝑍:ω–1-1→V)
89	3, 88	mpan2 707	1 ⊢ (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  ∪ cuni 4436  ∩ cint 4475   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115   ∘ ccom 5118  suc csuc 5725  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  ℩crio 6610  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065  seq_𝜔cseqom 7542   ↑_𝑚 cmap 7857   ≈ cen 7952  Fincfn 7955

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-rep 4771  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-3or 1038  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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by:  fin23lem32  9166