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Theorem funcnvuni 7285
Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 6119 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
funcnvuni (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)
Distinct variable group:   𝑓,𝑔,𝐴

Proof of Theorem funcnvuni
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnveq 5451 . . . . . . . 8 (𝑥 = 𝑣𝑥 = 𝑣)
21eqeq2d 2770 . . . . . . 7 (𝑥 = 𝑣 → (𝑧 = 𝑥𝑧 = 𝑣))
32cbvrexv 3311 . . . . . 6 (∃𝑥𝐴 𝑧 = 𝑥 ↔ ∃𝑣𝐴 𝑧 = 𝑣)
4 cnveq 5451 . . . . . . . . . . 11 (𝑓 = 𝑣𝑓 = 𝑣)
54funeqd 6071 . . . . . . . . . 10 (𝑓 = 𝑣 → (Fun 𝑓 ↔ Fun 𝑣))
6 sseq1 3767 . . . . . . . . . . . 12 (𝑓 = 𝑣 → (𝑓𝑔𝑣𝑔))
7 sseq2 3768 . . . . . . . . . . . 12 (𝑓 = 𝑣 → (𝑔𝑓𝑔𝑣))
86, 7orbi12d 748 . . . . . . . . . . 11 (𝑓 = 𝑣 → ((𝑓𝑔𝑔𝑓) ↔ (𝑣𝑔𝑔𝑣)))
98ralbidv 3124 . . . . . . . . . 10 (𝑓 = 𝑣 → (∀𝑔𝐴 (𝑓𝑔𝑔𝑓) ↔ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣)))
105, 9anbi12d 749 . . . . . . . . 9 (𝑓 = 𝑣 → ((Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) ↔ (Fun 𝑣 ∧ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣))))
1110rspcv 3445 . . . . . . . 8 (𝑣𝐴 → (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (Fun 𝑣 ∧ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣))))
12 funeq 6069 . . . . . . . . . 10 (𝑧 = 𝑣 → (Fun 𝑧 ↔ Fun 𝑣))
1312biimprcd 240 . . . . . . . . 9 (Fun 𝑣 → (𝑧 = 𝑣 → Fun 𝑧))
14 sseq2 3768 . . . . . . . . . . . . . . 15 (𝑔 = 𝑥 → (𝑣𝑔𝑣𝑥))
15 sseq1 3767 . . . . . . . . . . . . . . 15 (𝑔 = 𝑥 → (𝑔𝑣𝑥𝑣))
1614, 15orbi12d 748 . . . . . . . . . . . . . 14 (𝑔 = 𝑥 → ((𝑣𝑔𝑔𝑣) ↔ (𝑣𝑥𝑥𝑣)))
1716rspcv 3445 . . . . . . . . . . . . 13 (𝑥𝐴 → (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑣𝑥𝑥𝑣)))
18 cnvss 5450 . . . . . . . . . . . . . . . 16 (𝑣𝑥𝑣𝑥)
19 cnvss 5450 . . . . . . . . . . . . . . . 16 (𝑥𝑣𝑥𝑣)
2018, 19orim12i 539 . . . . . . . . . . . . . . 15 ((𝑣𝑥𝑥𝑣) → (𝑣𝑥𝑥𝑣))
21 sseq12 3769 . . . . . . . . . . . . . . . . 17 ((𝑧 = 𝑣𝑤 = 𝑥) → (𝑧𝑤𝑣𝑥))
2221ancoms 468 . . . . . . . . . . . . . . . 16 ((𝑤 = 𝑥𝑧 = 𝑣) → (𝑧𝑤𝑣𝑥))
23 sseq12 3769 . . . . . . . . . . . . . . . 16 ((𝑤 = 𝑥𝑧 = 𝑣) → (𝑤𝑧𝑥𝑣))
2422, 23orbi12d 748 . . . . . . . . . . . . . . 15 ((𝑤 = 𝑥𝑧 = 𝑣) → ((𝑧𝑤𝑤𝑧) ↔ (𝑣𝑥𝑥𝑣)))
2520, 24syl5ibrcom 237 . . . . . . . . . . . . . 14 ((𝑣𝑥𝑥𝑣) → ((𝑤 = 𝑥𝑧 = 𝑣) → (𝑧𝑤𝑤𝑧)))
2625expd 451 . . . . . . . . . . . . 13 ((𝑣𝑥𝑥𝑣) → (𝑤 = 𝑥 → (𝑧 = 𝑣 → (𝑧𝑤𝑤𝑧))))
2717, 26syl6com 37 . . . . . . . . . . . 12 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑥𝐴 → (𝑤 = 𝑥 → (𝑧 = 𝑣 → (𝑧𝑤𝑤𝑧)))))
2827rexlimdv 3168 . . . . . . . . . . 11 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (∃𝑥𝐴 𝑤 = 𝑥 → (𝑧 = 𝑣 → (𝑧𝑤𝑤𝑧))))
2928com23 86 . . . . . . . . . 10 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑧 = 𝑣 → (∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))
3029alrimdv 2006 . . . . . . . . 9 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑧 = 𝑣 → ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))
3113, 30anim12ii 595 . . . . . . . 8 ((Fun 𝑣 ∧ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣)) → (𝑧 = 𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
3211, 31syl6com 37 . . . . . . 7 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (𝑣𝐴 → (𝑧 = 𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))))
3332rexlimdv 3168 . . . . . 6 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (∃𝑣𝐴 𝑧 = 𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
343, 33syl5bi 232 . . . . 5 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (∃𝑥𝐴 𝑧 = 𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
3534alrimiv 2004 . . . 4 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → ∀𝑧(∃𝑥𝐴 𝑧 = 𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
36 df-ral 3055 . . . . 5 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧)) ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} → (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧))))
37 vex 3343 . . . . . . . 8 𝑧 ∈ V
38 eqeq1 2764 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 = 𝑥𝑧 = 𝑥))
3938rexbidv 3190 . . . . . . . 8 (𝑦 = 𝑧 → (∃𝑥𝐴 𝑦 = 𝑥 ↔ ∃𝑥𝐴 𝑧 = 𝑥))
4037, 39elab 3490 . . . . . . 7 (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} ↔ ∃𝑥𝐴 𝑧 = 𝑥)
41 eqeq1 2764 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦 = 𝑥𝑤 = 𝑥))
4241rexbidv 3190 . . . . . . . . 9 (𝑦 = 𝑤 → (∃𝑥𝐴 𝑦 = 𝑥 ↔ ∃𝑥𝐴 𝑤 = 𝑥))
4342ralab 3508 . . . . . . . 8 (∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧) ↔ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))
4443anbi2i 732 . . . . . . 7 ((Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧)) ↔ (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))
4540, 44imbi12i 339 . . . . . 6 ((𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} → (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧))) ↔ (∃𝑥𝐴 𝑧 = 𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
4645albii 1896 . . . . 5 (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} → (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧))) ↔ ∀𝑧(∃𝑥𝐴 𝑧 = 𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
4736, 46bitr2i 265 . . . 4 (∀𝑧(∃𝑥𝐴 𝑧 = 𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))) ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧)))
4835, 47sylib 208 . . 3 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → ∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧)))
49 fununi 6125 . . 3 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧)) → Fun {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥})
5048, 49syl 17 . 2 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥})
51 cnvuni 5464 . . . 4 𝐴 = 𝑥𝐴 𝑥
52 vex 3343 . . . . . 6 𝑥 ∈ V
5352cnvex 7279 . . . . 5 𝑥 ∈ V
5453dfiun2 4706 . . . 4 𝑥𝐴 𝑥 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥}
5551, 54eqtri 2782 . . 3 𝐴 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥}
5655funeqi 6070 . 2 (Fun 𝐴 ↔ Fun {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥})
5750, 56sylibr 224 1 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  wal 1630   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wrex 3051  wss 3715   cuni 4588   ciun 4672  ccnv 5265  Fun wfun 6043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-fun 6051
This theorem is referenced by:  fun11uni  7286
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