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Theorem funcnvuni 4968
Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 4960 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 4509 . . . . . . . 8 (𝑥 = 𝑣𝑥 = 𝑣)
21eqeq2d 2051 . . . . . . 7 (𝑥 = 𝑣 → (𝑧 = 𝑥𝑧 = 𝑣))
32cbvrexv 2534 . . . . . 6 (∃𝑥𝐴 𝑧 = 𝑥 ↔ ∃𝑣𝐴 𝑧 = 𝑣)
4 cnveq 4509 . . . . . . . . . . 11 (𝑓 = 𝑣𝑓 = 𝑣)
54funeqd 4923 . . . . . . . . . 10 (𝑓 = 𝑣 → (Fun 𝑓 ↔ Fun 𝑣))
6 sseq1 2966 . . . . . . . . . . . 12 (𝑓 = 𝑣 → (𝑓𝑔𝑣𝑔))
7 sseq2 2967 . . . . . . . . . . . 12 (𝑓 = 𝑣 → (𝑔𝑓𝑔𝑣))
86, 7orbi12d 707 . . . . . . . . . . 11 (𝑓 = 𝑣 → ((𝑓𝑔𝑔𝑓) ↔ (𝑣𝑔𝑔𝑣)))
98ralbidv 2326 . . . . . . . . . 10 (𝑓 = 𝑣 → (∀𝑔𝐴 (𝑓𝑔𝑔𝑓) ↔ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣)))
105, 9anbi12d 442 . . . . . . . . 9 (𝑓 = 𝑣 → ((Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) ↔ (Fun 𝑣 ∧ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣))))
1110rspcv 2652 . . . . . . . 8 (𝑣𝐴 → (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (Fun 𝑣 ∧ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣))))
12 funeq 4921 . . . . . . . . . 10 (𝑧 = 𝑣 → (Fun 𝑧 ↔ Fun 𝑣))
1312biimprcd 149 . . . . . . . . 9 (Fun 𝑣 → (𝑧 = 𝑣 → Fun 𝑧))
14 sseq2 2967 . . . . . . . . . . . . . . 15 (𝑔 = 𝑥 → (𝑣𝑔𝑣𝑥))
15 sseq1 2966 . . . . . . . . . . . . . . 15 (𝑔 = 𝑥 → (𝑔𝑣𝑥𝑣))
1614, 15orbi12d 707 . . . . . . . . . . . . . 14 (𝑔 = 𝑥 → ((𝑣𝑔𝑔𝑣) ↔ (𝑣𝑥𝑥𝑣)))
1716rspcv 2652 . . . . . . . . . . . . 13 (𝑥𝐴 → (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑣𝑥𝑥𝑣)))
18 cnvss 4508 . . . . . . . . . . . . . . . 16 (𝑣𝑥𝑣𝑥)
19 cnvss 4508 . . . . . . . . . . . . . . . 16 (𝑥𝑣𝑥𝑣)
2018, 19orim12i 676 . . . . . . . . . . . . . . 15 ((𝑣𝑥𝑥𝑣) → (𝑣𝑥𝑥𝑣))
21 sseq12 2968 . . . . . . . . . . . . . . . . 17 ((𝑧 = 𝑣𝑤 = 𝑥) → (𝑧𝑤𝑣𝑥))
2221ancoms 255 . . . . . . . . . . . . . . . 16 ((𝑤 = 𝑥𝑧 = 𝑣) → (𝑧𝑤𝑣𝑥))
23 sseq12 2968 . . . . . . . . . . . . . . . 16 ((𝑤 = 𝑥𝑧 = 𝑣) → (𝑤𝑧𝑥𝑣))
2422, 23orbi12d 707 . . . . . . . . . . . . . . 15 ((𝑤 = 𝑥𝑧 = 𝑣) → ((𝑧𝑤𝑤𝑧) ↔ (𝑣𝑥𝑥𝑣)))
2520, 24syl5ibrcom 146 . . . . . . . . . . . . . 14 ((𝑣𝑥𝑥𝑣) → ((𝑤 = 𝑥𝑧 = 𝑣) → (𝑧𝑤𝑤𝑧)))
2625expd 245 . . . . . . . . . . . . 13 ((𝑣𝑥𝑥𝑣) → (𝑤 = 𝑥 → (𝑧 = 𝑣 → (𝑧𝑤𝑤𝑧))))
2717, 26syl6com 31 . . . . . . . . . . . 12 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑥𝐴 → (𝑤 = 𝑥 → (𝑧 = 𝑣 → (𝑧𝑤𝑤𝑧)))))
2827rexlimdv 2432 . . . . . . . . . . 11 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (∃𝑥𝐴 𝑤 = 𝑥 → (𝑧 = 𝑣 → (𝑧𝑤𝑤𝑧))))
2928com23 72 . . . . . . . . . 10 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑧 = 𝑣 → (∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))
3029alrimdv 1756 . . . . . . . . 9 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑧 = 𝑣 → ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))
3113, 30anim12ii 325 . . . . . . . 8 ((Fun 𝑣 ∧ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣)) → (𝑧 = 𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
3211, 31syl6com 31 . . . . . . 7 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (𝑣𝐴 → (𝑧 = 𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))))
3332rexlimdv 2432 . . . . . 6 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (∃𝑣𝐴 𝑧 = 𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
343, 33syl5bi 141 . . . . 5 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (∃𝑥𝐴 𝑧 = 𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
3534alrimiv 1754 . . . 4 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → ∀𝑧(∃𝑥𝐴 𝑧 = 𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
36 df-ral 2311 . . . . 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 2560 . . . . . . . 8 𝑧 ∈ V
38 eqeq1 2046 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 = 𝑥𝑧 = 𝑥))
3938rexbidv 2327 . . . . . . . 8 (𝑦 = 𝑧 → (∃𝑥𝐴 𝑦 = 𝑥 ↔ ∃𝑥𝐴 𝑧 = 𝑥))
4037, 39elab 2687 . . . . . . 7 (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} ↔ ∃𝑥𝐴 𝑧 = 𝑥)
41 eqeq1 2046 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦 = 𝑥𝑤 = 𝑥))
4241rexbidv 2327 . . . . . . . . 9 (𝑦 = 𝑤 → (∃𝑥𝐴 𝑦 = 𝑥 ↔ ∃𝑥𝐴 𝑤 = 𝑥))
4342ralab 2701 . . . . . . . 8 (∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧) ↔ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))
4443anbi2i 430 . . . . . . 7 ((Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧)) ↔ (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))
4540, 44imbi12i 228 . . . . . 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 1359 . . . . 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 174 . . . 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 127 . . 3 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → ∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧)))
49 fununi 4967 . . 3 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧)) → Fun {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥})
5048, 49syl 14 . 2 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥})
51 cnvuni 4521 . . . 4 𝐴 = 𝑥𝐴 𝑥
52 vex 2560 . . . . . 6 𝑥 ∈ V
5352cnvex 4856 . . . . 5 𝑥 ∈ V
5453dfiun2 3691 . . . 4 𝑥𝐴 𝑥 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥}
5551, 54eqtri 2060 . . 3 𝐴 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥}
5655funeqi 4922 . 2 (Fun 𝐴 ↔ Fun {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥})
5750, 56sylibr 137 1 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wo 629  wal 1241   = wceq 1243  wcel 1393  {cab 2026  wral 2306  wrex 2307  wss 2917   cuni 3580   ciun 3657  ccnv 4344  Fun wfun 4896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904
This theorem is referenced by:  fun11uni  4969
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