MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fununi Structured version   Visualization version   GIF version

Theorem fununi 6104
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 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → 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.
StepHypRef Expression
1 funrel 6048 . . . . 5 (Fun 𝑓 → Rel 𝑓)
21adantr 466 . . . 4 ((Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Rel 𝑓)
32ralimi 3101 . . 3 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → ∀𝑓𝐴 Rel 𝑓)
4 reluni 5380 . . 3 (Rel 𝐴 ↔ ∀𝑓𝐴 Rel 𝑓)
53, 4sylibr 224 . 2 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Rel 𝐴)
6 r19.28v 3219 . . . 4 ((Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → ∀𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)))
76ralimi 3101 . . 3 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → ∀𝑓𝐴𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)))
8 ssel 3746 . . . . . . . . . . . 12 (𝑤𝑣 → (⟨𝑥, 𝑦⟩ ∈ 𝑤 → ⟨𝑥, 𝑦⟩ ∈ 𝑣))
98anim1d 598 . . . . . . . . . . 11 (𝑤𝑣 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝑣 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)))
10 dffun4 6043 . . . . . . . . . . . . . 14 (Fun 𝑣 ↔ (Rel 𝑣 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝑣 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
1110simprbi 484 . . . . . . . . . . . . 13 (Fun 𝑣 → ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝑣 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
121119.21bbi 2214 . . . . . . . . . . . 12 (Fun 𝑣 → ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝑣 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
131219.21bi 2213 . . . . . . . . . . 11 (Fun 𝑣 → ((⟨𝑥, 𝑦⟩ ∈ 𝑣 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
149, 13syl9r 78 . . . . . . . . . 10 (Fun 𝑣 → (𝑤𝑣 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
1514adantl 467 . . . . . . . . 9 ((Fun 𝑤 ∧ Fun 𝑣) → (𝑤𝑣 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
16 ssel 3746 . . . . . . . . . . . 12 (𝑣𝑤 → (⟨𝑥, 𝑧⟩ ∈ 𝑣 → ⟨𝑥, 𝑧⟩ ∈ 𝑤))
1716anim2d 599 . . . . . . . . . . 11 (𝑣𝑤 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑤)))
18 dffun4 6043 . . . . . . . . . . . . . 14 (Fun 𝑤 ↔ (Rel 𝑤 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑤) → 𝑦 = 𝑧)))
1918simprbi 484 . . . . . . . . . . . . 13 (Fun 𝑤 → ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑤) → 𝑦 = 𝑧))
201919.21bbi 2214 . . . . . . . . . . . 12 (Fun 𝑤 → ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑤) → 𝑦 = 𝑧))
212019.21bi 2213 . . . . . . . . . . 11 (Fun 𝑤 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑤) → 𝑦 = 𝑧))
2217, 21syl9r 78 . . . . . . . . . 10 (Fun 𝑤 → (𝑣𝑤 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
2322adantr 466 . . . . . . . . 9 ((Fun 𝑤 ∧ Fun 𝑣) → (𝑣𝑤 → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
2415, 23jaod 848 . . . . . . . 8 ((Fun 𝑤 ∧ Fun 𝑣) → ((𝑤𝑣𝑣𝑤) → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
2524imp 393 . . . . . . 7 (((Fun 𝑤 ∧ Fun 𝑣) ∧ (𝑤𝑣𝑣𝑤)) → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
26252ralimi 3102 . . . . . 6 (∀𝑤𝐴𝑣𝐴 ((Fun 𝑤 ∧ Fun 𝑣) ∧ (𝑤𝑣𝑣𝑤)) → ∀𝑤𝐴𝑣𝐴 ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
27 funeq 6051 . . . . . . . . . 10 (𝑓 = 𝑤 → (Fun 𝑓 ↔ Fun 𝑤))
28 sseq1 3775 . . . . . . . . . . 11 (𝑓 = 𝑤 → (𝑓𝑔𝑤𝑔))
29 sseq2 3776 . . . . . . . . . . 11 (𝑓 = 𝑤 → (𝑔𝑓𝑔𝑤))
3028, 29orbi12d 904 . . . . . . . . . 10 (𝑓 = 𝑤 → ((𝑓𝑔𝑔𝑓) ↔ (𝑤𝑔𝑔𝑤)))
3127, 30anbi12d 616 . . . . . . . . 9 (𝑓 = 𝑤 → ((Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)) ↔ (Fun 𝑤 ∧ (𝑤𝑔𝑔𝑤))))
32 sseq2 3776 . . . . . . . . . . 11 (𝑔 = 𝑣 → (𝑤𝑔𝑤𝑣))
33 sseq1 3775 . . . . . . . . . . 11 (𝑔 = 𝑣 → (𝑔𝑤𝑣𝑤))
3432, 33orbi12d 904 . . . . . . . . . 10 (𝑔 = 𝑣 → ((𝑤𝑔𝑔𝑤) ↔ (𝑤𝑣𝑣𝑤)))
3534anbi2d 614 . . . . . . . . 9 (𝑔 = 𝑣 → ((Fun 𝑤 ∧ (𝑤𝑔𝑔𝑤)) ↔ (Fun 𝑤 ∧ (𝑤𝑣𝑣𝑤))))
3631, 35cbvral2v 3328 . . . . . . . 8 (∀𝑓𝐴𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)) ↔ ∀𝑤𝐴𝑣𝐴 (Fun 𝑤 ∧ (𝑤𝑣𝑣𝑤)))
37 ralcom 3246 . . . . . . . . 9 (∀𝑓𝐴𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)) ↔ ∀𝑔𝐴𝑓𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)))
38 orcom 859 . . . . . . . . . . . 12 ((𝑓𝑔𝑔𝑓) ↔ (𝑔𝑓𝑓𝑔))
39 sseq1 3775 . . . . . . . . . . . . 13 (𝑔 = 𝑤 → (𝑔𝑓𝑤𝑓))
40 sseq2 3776 . . . . . . . . . . . . 13 (𝑔 = 𝑤 → (𝑓𝑔𝑓𝑤))
4139, 40orbi12d 904 . . . . . . . . . . . 12 (𝑔 = 𝑤 → ((𝑔𝑓𝑓𝑔) ↔ (𝑤𝑓𝑓𝑤)))
4238, 41syl5bb 272 . . . . . . . . . . 11 (𝑔 = 𝑤 → ((𝑓𝑔𝑔𝑓) ↔ (𝑤𝑓𝑓𝑤)))
4342anbi2d 614 . . . . . . . . . 10 (𝑔 = 𝑤 → ((Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)) ↔ (Fun 𝑓 ∧ (𝑤𝑓𝑓𝑤))))
44 funeq 6051 . . . . . . . . . . 11 (𝑓 = 𝑣 → (Fun 𝑓 ↔ Fun 𝑣))
45 sseq2 3776 . . . . . . . . . . . 12 (𝑓 = 𝑣 → (𝑤𝑓𝑤𝑣))
46 sseq1 3775 . . . . . . . . . . . 12 (𝑓 = 𝑣 → (𝑓𝑤𝑣𝑤))
4745, 46orbi12d 904 . . . . . . . . . . 11 (𝑓 = 𝑣 → ((𝑤𝑓𝑓𝑤) ↔ (𝑤𝑣𝑣𝑤)))
4844, 47anbi12d 616 . . . . . . . . . 10 (𝑓 = 𝑣 → ((Fun 𝑓 ∧ (𝑤𝑓𝑓𝑤)) ↔ (Fun 𝑣 ∧ (𝑤𝑣𝑣𝑤))))
4943, 48cbvral2v 3328 . . . . . . . . 9 (∀𝑔𝐴𝑓𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)) ↔ ∀𝑤𝐴𝑣𝐴 (Fun 𝑣 ∧ (𝑤𝑣𝑣𝑤)))
5037, 49bitri 264 . . . . . . . 8 (∀𝑓𝐴𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)) ↔ ∀𝑤𝐴𝑣𝐴 (Fun 𝑣 ∧ (𝑤𝑣𝑣𝑤)))
5136, 50anbi12i 612 . . . . . . 7 ((∀𝑓𝐴𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)) ∧ ∀𝑓𝐴𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓))) ↔ (∀𝑤𝐴𝑣𝐴 (Fun 𝑤 ∧ (𝑤𝑣𝑣𝑤)) ∧ ∀𝑤𝐴𝑣𝐴 (Fun 𝑣 ∧ (𝑤𝑣𝑣𝑤))))
52 anidm 554 . . . . . . 7 ((∀𝑓𝐴𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)) ∧ ∀𝑓𝐴𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓))) ↔ ∀𝑓𝐴𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)))
53 anandir 656 . . . . . . . . 9 (((Fun 𝑤 ∧ Fun 𝑣) ∧ (𝑤𝑣𝑣𝑤)) ↔ ((Fun 𝑤 ∧ (𝑤𝑣𝑣𝑤)) ∧ (Fun 𝑣 ∧ (𝑤𝑣𝑣𝑤))))
54532ralbii 3130 . . . . . . . 8 (∀𝑤𝐴𝑣𝐴 ((Fun 𝑤 ∧ Fun 𝑣) ∧ (𝑤𝑣𝑣𝑤)) ↔ ∀𝑤𝐴𝑣𝐴 ((Fun 𝑤 ∧ (𝑤𝑣𝑣𝑤)) ∧ (Fun 𝑣 ∧ (𝑤𝑣𝑣𝑤))))
55 r19.26-2 3213 . . . . . . . 8 (∀𝑤𝐴𝑣𝐴 ((Fun 𝑤 ∧ (𝑤𝑣𝑣𝑤)) ∧ (Fun 𝑣 ∧ (𝑤𝑣𝑣𝑤))) ↔ (∀𝑤𝐴𝑣𝐴 (Fun 𝑤 ∧ (𝑤𝑣𝑣𝑤)) ∧ ∀𝑤𝐴𝑣𝐴 (Fun 𝑣 ∧ (𝑤𝑣𝑣𝑤))))
5654, 55bitr2i 265 . . . . . . 7 ((∀𝑤𝐴𝑣𝐴 (Fun 𝑤 ∧ (𝑤𝑣𝑣𝑤)) ∧ ∀𝑤𝐴𝑣𝐴 (Fun 𝑣 ∧ (𝑤𝑣𝑣𝑤))) ↔ ∀𝑤𝐴𝑣𝐴 ((Fun 𝑤 ∧ Fun 𝑣) ∧ (𝑤𝑣𝑣𝑤)))
5751, 52, 563bitr3i 290 . . . . . 6 (∀𝑓𝐴𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)) ↔ ∀𝑤𝐴𝑣𝐴 ((Fun 𝑤 ∧ Fun 𝑣) ∧ (𝑤𝑣𝑣𝑤)))
58 eluni 4577 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑤(⟨𝑥, 𝑦⟩ ∈ 𝑤𝑤𝐴))
59 eluni 4577 . . . . . . . . . 10 (⟨𝑥, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑣(⟨𝑥, 𝑧⟩ ∈ 𝑣𝑣𝐴))
6058, 59anbi12i 612 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ (∃𝑤(⟨𝑥, 𝑦⟩ ∈ 𝑤𝑤𝐴) ∧ ∃𝑣(⟨𝑥, 𝑧⟩ ∈ 𝑣𝑣𝐴)))
61 eeanv 2344 . . . . . . . . 9 (∃𝑤𝑣((⟨𝑥, 𝑦⟩ ∈ 𝑤𝑤𝐴) ∧ (⟨𝑥, 𝑧⟩ ∈ 𝑣𝑣𝐴)) ↔ (∃𝑤(⟨𝑥, 𝑦⟩ ∈ 𝑤𝑤𝐴) ∧ ∃𝑣(⟨𝑥, 𝑧⟩ ∈ 𝑣𝑣𝐴)))
62 an4 635 . . . . . . . . . . 11 (((⟨𝑥, 𝑦⟩ ∈ 𝑤𝑤𝐴) ∧ (⟨𝑥, 𝑧⟩ ∈ 𝑣𝑣𝐴)) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) ∧ (𝑤𝐴𝑣𝐴)))
63 ancom 452 . . . . . . . . . . 11 (((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) ∧ (𝑤𝐴𝑣𝐴)) ↔ ((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)))
6462, 63bitri 264 . . . . . . . . . 10 (((⟨𝑥, 𝑦⟩ ∈ 𝑤𝑤𝐴) ∧ (⟨𝑥, 𝑧⟩ ∈ 𝑣𝑣𝐴)) ↔ ((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)))
65642exbii 1925 . . . . . . . . 9 (∃𝑤𝑣((⟨𝑥, 𝑦⟩ ∈ 𝑤𝑤𝐴) ∧ (⟨𝑥, 𝑧⟩ ∈ 𝑣𝑣𝐴)) ↔ ∃𝑤𝑣((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)))
6660, 61, 653bitr2i 288 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ ∃𝑤𝑣((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)))
6766imbi1i 338 . . . . . . 7 (((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧) ↔ (∃𝑤𝑣((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧))
68 19.23v 2023 . . . . . . 7 (∀𝑤(∃𝑣((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧) ↔ (∃𝑤𝑣((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧))
69 r2al 3088 . . . . . . . 8 (∀𝑤𝐴𝑣𝐴 ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧) ↔ ∀𝑤𝑣((𝑤𝐴𝑣𝐴) → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
70 impexp 437 . . . . . . . . 9 ((((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧) ↔ ((𝑤𝐴𝑣𝐴) → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
71702albii 1896 . . . . . . . 8 (∀𝑤𝑣(((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧) ↔ ∀𝑤𝑣((𝑤𝐴𝑣𝐴) → ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧)))
72 19.23v 2023 . . . . . . . . 9 (∀𝑣(((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧) ↔ (∃𝑣((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧))
7372albii 1895 . . . . . . . 8 (∀𝑤𝑣(((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧) ↔ ∀𝑤(∃𝑣((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧))
7469, 71, 733bitr2ri 289 . . . . . . 7 (∀𝑤(∃𝑣((𝑤𝐴𝑣𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣)) → 𝑦 = 𝑧) ↔ ∀𝑤𝐴𝑣𝐴 ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
7567, 68, 743bitr2i 288 . . . . . 6 (((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧) ↔ ∀𝑤𝐴𝑣𝐴 ((⟨𝑥, 𝑦⟩ ∈ 𝑤 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑣) → 𝑦 = 𝑧))
7626, 57, 753imtr4i 281 . . . . 5 (∀𝑓𝐴𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)) → ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
7776alrimiv 2007 . . . 4 (∀𝑓𝐴𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)) → ∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
7877alrimivv 2008 . . 3 (∀𝑓𝐴𝑔𝐴 (Fun 𝑓 ∧ (𝑓𝑔𝑔𝑓)) → ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
797, 78syl 17 . 2 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
80 dffun4 6043 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧)))
815, 79, 80sylanbrc 572 1 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 836  wal 1629  wex 1852  wcel 2145  wral 3061  wss 3723  cop 4322   cuni 4574  Rel wrel 5254  Fun wfun 6025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-id 5157  df-rel 5256  df-cnv 5257  df-co 5258  df-fun 6033
This theorem is referenced by:  funcnvuni  7266  fun11uni  7267  axdc3lem2  9475
  Copyright terms: Public domain W3C validator