Theorem fiun 7885

Description: The union of a chain (with respect to inclusion) of functions is a function. Analogous to f1iun 7886. (Contributed by AV, 6-Oct-2023.)

Hypotheses

Ref	Expression
fiun.1	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
fiun.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
fiun	⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵:∪ 𝑥 ∈ 𝐴 𝐷⟶𝑆)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶   𝑥,𝑦   𝑥,𝑆

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝑆(𝑦)

Proof of Theorem fiun

Dummy variables 𝑣 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3442	. . . . . . . 8 ⊢ 𝑢 ∈ V
2		eqeq1 2738	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → (𝑧 = 𝐵 ↔ 𝑢 = 𝐵))
3	2	rexbidv 3158	. . . . . . . 8 ⊢ (𝑧 = 𝑢 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑢 = 𝐵))
4	1, 3	elab 3632	. . . . . . 7 ⊢ (𝑢 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑢 = 𝐵)
5		r19.29 3097	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ ∃𝑥 ∈ 𝐴 𝑢 = 𝐵) → ∃𝑥 ∈ 𝐴 ((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 = 𝐵))
6		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑥Fun 𝑢
7		nfre1 3259	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑧 = 𝐵
8	7	nfab 2902	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}
9		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)
10	8, 9	nfralw 3281	. . . . . . . . . 10 ⊢ Ⅎ𝑥∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)
11	6, 10	nfan 1900	. . . . . . . . 9 ⊢ Ⅎ𝑥(Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
12		ffun 6663	. . . . . . . . . . . . 13 ⊢ (𝐵:𝐷⟶𝑆 → Fun 𝐵)
13		funeq 6510	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝐵 → (Fun 𝑢 ↔ Fun 𝐵))
14		bianir 1058	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐵 ∧ (Fun 𝑢 ↔ Fun 𝐵)) → Fun 𝑢)
15	12, 13, 14	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝐵:𝐷⟶𝑆 ∧ 𝑢 = 𝐵) → Fun 𝑢)
16	15	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 = 𝐵) → Fun 𝑢)
17		fiun.1	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
18	17	fiunlem 7884	. . . . . . . . . . 11 ⊢ (((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
19	16, 18	jca 511	. . . . . . . . . 10 ⊢ (((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 = 𝐵) → (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)))
20	19	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 = 𝐵) → (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))))
21	11, 20	rexlimi 3234	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 = 𝐵) → (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)))
22	5, 21	syl 17	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ ∃𝑥 ∈ 𝐴 𝑢 = 𝐵) → (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)))
23	4, 22	sylan2b 594	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}) → (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)))
24	23	ralrimiva 3126	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ∀𝑢 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)))
25		fununi 6565	. . . . 5 ⊢ (∀𝑢 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)) → Fun ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵})
26	24, 25	syl 17	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → Fun ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵})
27		fiun.2	. . . . . 6 ⊢ 𝐵 ∈ V
28	27	dfiun2 4985	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}
29	28	funeqi 6511	. . . 4 ⊢ (Fun ∪ 𝑥 ∈ 𝐴 𝐵 ↔ Fun ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵})
30	26, 29	sylibr 234	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → Fun ∪ 𝑥 ∈ 𝐴 𝐵)
31	1	eldm2 5848	. . . . . . . 8 ⊢ (𝑢 ∈ dom 𝐵 ↔ ∃𝑣⟨𝑢, 𝑣⟩ ∈ 𝐵)
32		fdm 6669	. . . . . . . . 9 ⊢ (𝐵:𝐷⟶𝑆 → dom 𝐵 = 𝐷)
33	32	eleq2d 2820	. . . . . . . 8 ⊢ (𝐵:𝐷⟶𝑆 → (𝑢 ∈ dom 𝐵 ↔ 𝑢 ∈ 𝐷))
34	31, 33	bitr3id 285	. . . . . . 7 ⊢ (𝐵:𝐷⟶𝑆 → (∃𝑣⟨𝑢, 𝑣⟩ ∈ 𝐵 ↔ 𝑢 ∈ 𝐷))
35	34	adantr 480	. . . . . 6 ⊢ ((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → (∃𝑣⟨𝑢, 𝑣⟩ ∈ 𝐵 ↔ 𝑢 ∈ 𝐷))
36	35	ralrexbid 3091	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → (∃𝑥 ∈ 𝐴 ∃𝑣⟨𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑢 ∈ 𝐷))
37		eliun 4948	. . . . . . 7 ⊢ (⟨𝑢, 𝑣⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑢, 𝑣⟩ ∈ 𝐵)
38	37	exbii 1849	. . . . . 6 ⊢ (∃𝑣⟨𝑢, 𝑣⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑣∃𝑥 ∈ 𝐴 ⟨𝑢, 𝑣⟩ ∈ 𝐵)
39	1	eldm2 5848	. . . . . 6 ⊢ (𝑢 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑣⟨𝑢, 𝑣⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
40		rexcom4 3261	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑣⟨𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑣∃𝑥 ∈ 𝐴 ⟨𝑢, 𝑣⟩ ∈ 𝐵)
41	38, 39, 40	3bitr4i 303	. . . . 5 ⊢ (𝑢 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑣⟨𝑢, 𝑣⟩ ∈ 𝐵)
42		eliun 4948	. . . . 5 ⊢ (𝑢 ∈ ∪ 𝑥 ∈ 𝐴 𝐷 ↔ ∃𝑥 ∈ 𝐴 𝑢 ∈ 𝐷)
43	36, 41, 42	3bitr4g 314	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → (𝑢 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑢 ∈ ∪ 𝑥 ∈ 𝐴 𝐷))
44	43	eqrdv 2732	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐷)
45		df-fn 6493	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 Fn ∪ 𝑥 ∈ 𝐴 𝐷 ↔ (Fun ∪ 𝑥 ∈ 𝐴 𝐵 ∧ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐷))
46	30, 44, 45	sylanbrc 583	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵 Fn ∪ 𝑥 ∈ 𝐴 𝐷)
47		rniun 6103	. . 3 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵
48		frn 6667	. . . . . 6 ⊢ (𝐵:𝐷⟶𝑆 → ran 𝐵 ⊆ 𝑆)
49	48	adantr 480	. . . . 5 ⊢ ((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ran 𝐵 ⊆ 𝑆)
50	49	ralimi 3071	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ∀𝑥 ∈ 𝐴 ran 𝐵 ⊆ 𝑆)
51		iunss 4998	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 ran 𝐵 ⊆ 𝑆 ↔ ∀𝑥 ∈ 𝐴 ran 𝐵 ⊆ 𝑆)
52	50, 51	sylibr 234	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ∪ 𝑥 ∈ 𝐴 ran 𝐵 ⊆ 𝑆)
53	47, 52	eqsstrid 3970	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ran ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑆)
54		df-f 6494	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵:∪ 𝑥 ∈ 𝐴 𝐷⟶𝑆 ↔ (∪ 𝑥 ∈ 𝐴 𝐵 Fn ∪ 𝑥 ∈ 𝐴 𝐷 ∧ ran ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑆))
55	46, 53, 54	sylanbrc 583	1 ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵:∪ 𝑥 ∈ 𝐴 𝐷⟶𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2712  ∀wral 3049  ∃wrex 3058  Vcvv 3438   ⊆ wss 3899  ⟨cop 4584  ∪ cuni 4861  ∪ ciun 4944  dom cdm 5622  ran crn 5623  Fun wfun 6484   Fn wfn 6485  ⟶wf 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-fun 6492  df-fn 6493  df-f 6494

This theorem is referenced by:  satfun  35554