Theorem fununi 6455

Description: The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)

Assertion

Ref	Expression
fununi	⊢ (∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ∪ 𝐴)

Distinct variable group:   𝑓,𝑔,𝐴

Proof of Theorem fununi

Dummy variables 𝑥 𝑦 𝑧 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funrel 6397	. . . . 5 ⊢ (Fun 𝑓 → Rel 𝑓)
2	1	adantr 484	. . . 4 ⊢ ((Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Rel 𝑓)
3	2	ralimi 3083	. . 3 ⊢ (∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → ∀𝑓 ∈ 𝐴 Rel 𝑓)
4		reluni 5688	. . 3 ⊢ (Rel ∪ 𝐴 ↔ ∀𝑓 ∈ 𝐴 Rel 𝑓)
5	3, 4	sylibr 237	. 2 ⊢ (∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Rel ∪ 𝐴)
6		r19.28v 3108	. . . 4 ⊢ ((Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)))
7	6	ralimi 3083	. . 3 ⊢ (∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)))
8		ssel 3893	. . . . . . . . . . . 12 ⊢ (𝑤 ⊆ 𝑣 → (⟨𝑥, 𝑦⟩ ∈ 𝑤 → ⟨𝑥, 𝑦⟩ ∈ 𝑣))
9	8	anim1d 614	. . . . . . . . . . 11 ⊢ (𝑤 ⊆ 𝑣 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝑣 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)))
10		dffun4 6392	. . . . . . . . . . . . . 14 ⊢ (Fun 𝑣 ↔ (Rel 𝑣 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝑣 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
11	10	simprbi 500	. . . . . . . . . . . . 13 ⊢ (Fun 𝑣 → ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝑣 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
12	11	19.21bbi 2187	. . . . . . . . . . . 12 ⊢ (Fun 𝑣 → ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝑣 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
13	12	19.21bi 2186	. . . . . . . . . . 11 ⊢ (Fun 𝑣 → ((⟨𝑥, 𝑦⟩ ∈ 𝑣 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
14	9, 13	syl9r 78	. . . . . . . . . 10 ⊢ (Fun 𝑣 → (𝑤 ⊆ 𝑣 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
15	14	adantl 485	. . . . . . . . 9 ⊢ ((Fun 𝑤 ∧ Fun 𝑣) → (𝑤 ⊆ 𝑣 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
16		ssel 3893	. . . . . . . . . . . 12 ⊢ (𝑣 ⊆ 𝑤 → (⟨𝑥, 𝑧⟩ ∈ 𝑣 → ⟨𝑥, 𝑧⟩ ∈ 𝑤))
17	16	anim2d 615	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ 𝑤 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑤)))
18		dffun4 6392	. . . . . . . . . . . . . 14 ⊢ (Fun 𝑤 ↔ (Rel 𝑤 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑤) → 𝑦 = 𝑧)))
19	18	simprbi 500	. . . . . . . . . . . . 13 ⊢ (Fun 𝑤 → ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑤) → 𝑦 = 𝑧))
20	19	19.21bbi 2187	. . . . . . . . . . . 12 ⊢ (Fun 𝑤 → ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑤) → 𝑦 = 𝑧))
21	20	19.21bi 2186	. . . . . . . . . . 11 ⊢ (Fun 𝑤 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑤) → 𝑦 = 𝑧))
22	17, 21	syl9r 78	. . . . . . . . . 10 ⊢ (Fun 𝑤 → (𝑣 ⊆ 𝑤 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
23	22	adantr 484	. . . . . . . . 9 ⊢ ((Fun 𝑤 ∧ Fun 𝑣) → (𝑣 ⊆ 𝑤 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
24	15, 23	jaod 859	. . . . . . . 8 ⊢ ((Fun 𝑤 ∧ Fun 𝑣) → ((𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤) → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
25	24	imp 410	. . . . . . 7 ⊢ (((Fun 𝑤 ∧ Fun 𝑣) ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)) → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
26	25	2ralimi 3084	. . . . . 6 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((Fun 𝑤 ∧ Fun 𝑣) ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)) → ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
27		funeq 6400	. . . . . . . . . 10 ⊢ (𝑓 = 𝑤 → (Fun 𝑓 ↔ Fun 𝑤))
28		sseq1 3926	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑤 → (𝑓 ⊆ 𝑔 ↔ 𝑤 ⊆ 𝑔))
29		sseq2 3927	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑤 → (𝑔 ⊆ 𝑓 ↔ 𝑔 ⊆ 𝑤))
30	28, 29	orbi12d 919	. . . . . . . . . 10 ⊢ (𝑓 = 𝑤 → ((𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓) ↔ (𝑤 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑤)))
31	27, 30	anbi12d 634	. . . . . . . . 9 ⊢ (𝑓 = 𝑤 → ((Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) ↔ (Fun 𝑤 ∧ (𝑤 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑤))))
32		sseq2 3927	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑣 → (𝑤 ⊆ 𝑔 ↔ 𝑤 ⊆ 𝑣))
33		sseq1 3926	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑣 → (𝑔 ⊆ 𝑤 ↔ 𝑣 ⊆ 𝑤))
34	32, 33	orbi12d 919	. . . . . . . . . 10 ⊢ (𝑔 = 𝑣 → ((𝑤 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑤) ↔ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)))
35	34	anbi2d 632	. . . . . . . . 9 ⊢ (𝑔 = 𝑣 → ((Fun 𝑤 ∧ (𝑤 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑤)) ↔ (Fun 𝑤 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤))))
36	31, 35	cbvral2vw 3371	. . . . . . . 8 ⊢ (∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (Fun 𝑤 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)))
37		ralcom 3267	. . . . . . . . 9 ⊢ (∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) ↔ ∀𝑔 ∈ 𝐴 ∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)))
38		orcom 870	. . . . . . . . . . . 12 ⊢ ((𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓) ↔ (𝑔 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑔))
39		sseq1 3926	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑤 → (𝑔 ⊆ 𝑓 ↔ 𝑤 ⊆ 𝑓))
40		sseq2 3927	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑤 → (𝑓 ⊆ 𝑔 ↔ 𝑓 ⊆ 𝑤))
41	39, 40	orbi12d 919	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑤 → ((𝑔 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑔) ↔ (𝑤 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑤)))
42	38, 41	syl5bb 286	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑤 → ((𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓) ↔ (𝑤 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑤)))
43	42	anbi2d 632	. . . . . . . . . 10 ⊢ (𝑔 = 𝑤 → ((Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) ↔ (Fun 𝑓 ∧ (𝑤 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑤))))
44		funeq 6400	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑣 → (Fun 𝑓 ↔ Fun 𝑣))
45		sseq2 3927	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑣 → (𝑤 ⊆ 𝑓 ↔ 𝑤 ⊆ 𝑣))
46		sseq1 3926	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑣 → (𝑓 ⊆ 𝑤 ↔ 𝑣 ⊆ 𝑤))
47	45, 46	orbi12d 919	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑣 → ((𝑤 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑤) ↔ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)))
48	44, 47	anbi12d 634	. . . . . . . . . 10 ⊢ (𝑓 = 𝑣 → ((Fun 𝑓 ∧ (𝑤 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑤)) ↔ (Fun 𝑣 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤))))
49	43, 48	cbvral2vw 3371	. . . . . . . . 9 ⊢ (∀𝑔 ∈ 𝐴 ∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (Fun 𝑣 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)))
50	37, 49	bitri 278	. . . . . . . 8 ⊢ (∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (Fun 𝑣 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)))
51	36, 50	anbi12i 630	. . . . . . 7 ⊢ ((∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓))) ↔ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (Fun 𝑤 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)) ∧ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (Fun 𝑣 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤))))
52		anidm 568	. . . . . . 7 ⊢ ((∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓))) ↔ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)))
53		anandir 677	. . . . . . . . 9 ⊢ (((Fun 𝑤 ∧ Fun 𝑣) ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)) ↔ ((Fun 𝑤 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)) ∧ (Fun 𝑣 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤))))
54	53	2ralbii 3089	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((Fun 𝑤 ∧ Fun 𝑣) ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)) ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((Fun 𝑤 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)) ∧ (Fun 𝑣 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤))))
55		r19.26-2 3093	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((Fun 𝑤 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)) ∧ (Fun 𝑣 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤))) ↔ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (Fun 𝑤 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)) ∧ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (Fun 𝑣 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤))))
56	54, 55	bitr2i 279	. . . . . . 7 ⊢ ((∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (Fun 𝑤 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)) ∧ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (Fun 𝑣 ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤))) ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((Fun 𝑤 ∧ Fun 𝑣) ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)))
57	51, 52, 56	3bitr3i 304	. . . . . 6 ⊢ (∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((Fun 𝑤 ∧ Fun 𝑣) ∧ (𝑤 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑤)))
58		eluni 4822	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ↔ ∃𝑤(⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
59		eluni 4822	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑣(⟨𝑥, 𝑧⟩ ∈ 𝑣 ∧ 𝑣 ∈ 𝐴))
60	58, 59	anbi12i 630	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴) ↔ (∃𝑤(⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ 𝑤 ∈ 𝐴) ∧ ∃𝑣(⟨𝑥, 𝑧⟩ ∈ 𝑣 ∧ 𝑣 ∈ 𝐴)))
61		exdistrv 1964	. . . . . . . . 9 ⊢ (∃𝑤∃𝑣((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ 𝑤 ∈ 𝐴) ∧ (⟨𝑥, 𝑧⟩ ∈ 𝑣 ∧ 𝑣 ∈ 𝐴)) ↔ (∃𝑤(⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ 𝑤 ∈ 𝐴) ∧ ∃𝑣(⟨𝑥, 𝑧⟩ ∈ 𝑣 ∧ 𝑣 ∈ 𝐴)))
62		an4 656	. . . . . . . . . . 11 ⊢ (((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ 𝑤 ∈ 𝐴) ∧ (⟨𝑥, 𝑧⟩ ∈ 𝑣 ∧ 𝑣 ∈ 𝐴)) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) ∧ (𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)))
63	62	biancomi 466	. . . . . . . . . 10 ⊢ (((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ 𝑤 ∈ 𝐴) ∧ (⟨𝑥, 𝑧⟩ ∈ 𝑣 ∧ 𝑣 ∈ 𝐴)) ↔ ((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)))
64	63	2exbii 1856	. . . . . . . . 9 ⊢ (∃𝑤∃𝑣((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ 𝑤 ∈ 𝐴) ∧ (⟨𝑥, 𝑧⟩ ∈ 𝑣 ∧ 𝑣 ∈ 𝐴)) ↔ ∃𝑤∃𝑣((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)))
65	60, 61, 64	3bitr2i 302	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴) ↔ ∃𝑤∃𝑣((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)))
66	65	imbi1i 353	. . . . . . 7 ⊢ (((⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴) → 𝑦 = 𝑧) ↔ (∃𝑤∃𝑣((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧))
67		19.23v 1950	. . . . . . 7 ⊢ (∀𝑤(∃𝑣((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧) ↔ (∃𝑤∃𝑣((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧))
68		r2al 3122	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧) ↔ ∀𝑤∀𝑣((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
69		impexp 454	. . . . . . . . 9 ⊢ ((((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧) ↔ ((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
70	69	2albii 1828	. . . . . . . 8 ⊢ (∀𝑤∀𝑣(((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧) ↔ ∀𝑤∀𝑣((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
71		19.23v 1950	. . . . . . . . 9 ⊢ (∀𝑣(((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧) ↔ (∃𝑣((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧))
72	71	albii 1827	. . . . . . . 8 ⊢ (∀𝑤∀𝑣(((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧) ↔ ∀𝑤(∃𝑣((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧))
73	68, 70, 72	3bitr2ri 303	. . . . . . 7 ⊢ (∀𝑤(∃𝑣((𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧) ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
74	66, 67, 73	3bitr2i 302	. . . . . 6 ⊢ (((⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴) → 𝑦 = 𝑧) ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
75	26, 57, 74	3imtr4i 295	. . . . 5 ⊢ (∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → ((⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴) → 𝑦 = 𝑧))
76	75	alrimiv 1935	. . . 4 ⊢ (∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → ∀𝑧((⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴) → 𝑦 = 𝑧))
77	76	alrimivv 1936	. . 3 ⊢ (∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (Fun 𝑓 ∧ (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴) → 𝑦 = 𝑧))
78	7, 77	syl 17	. 2 ⊢ (∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴) → 𝑦 = 𝑧))
79		dffun4 6392	. 2 ⊢ (Fun ∪ 𝐴 ↔ (Rel ∪ 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴) → 𝑦 = 𝑧)))
80	5, 78, 79	sylanbrc 586	1 ⊢ (∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 847  ∀wal 1541  ∃wex 1787   ∈ wcel 2110  ∀wral 3061   ⊆ wss 3866  ⟨cop 4547  ∪ cuni 4819  Rel wrel 5556  Fun wfun 6374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-id 5455  df-rel 5558  df-cnv 5559  df-co 5560  df-fun 6382

This theorem is referenced by:  funcnvuni  7709  fun11uni  7710  fiun  7716  axdc3lem2  10065