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Theorem funcnvuni 5150
 Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5142 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 4673 . . . . . . . 8 (𝑥 = 𝑣𝑥 = 𝑣)
21eqeq2d 2126 . . . . . . 7 (𝑥 = 𝑣 → (𝑧 = 𝑥𝑧 = 𝑣))
32cbvrexv 2629 . . . . . 6 (∃𝑥𝐴 𝑧 = 𝑥 ↔ ∃𝑣𝐴 𝑧 = 𝑣)
4 cnveq 4673 . . . . . . . . . . 11 (𝑓 = 𝑣𝑓 = 𝑣)
54funeqd 5103 . . . . . . . . . 10 (𝑓 = 𝑣 → (Fun 𝑓 ↔ Fun 𝑣))
6 sseq1 3086 . . . . . . . . . . . 12 (𝑓 = 𝑣 → (𝑓𝑔𝑣𝑔))
7 sseq2 3087 . . . . . . . . . . . 12 (𝑓 = 𝑣 → (𝑔𝑓𝑔𝑣))
86, 7orbi12d 765 . . . . . . . . . . 11 (𝑓 = 𝑣 → ((𝑓𝑔𝑔𝑓) ↔ (𝑣𝑔𝑔𝑣)))
98ralbidv 2411 . . . . . . . . . 10 (𝑓 = 𝑣 → (∀𝑔𝐴 (𝑓𝑔𝑔𝑓) ↔ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣)))
105, 9anbi12d 462 . . . . . . . . 9 (𝑓 = 𝑣 → ((Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) ↔ (Fun 𝑣 ∧ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣))))
1110rspcv 2756 . . . . . . . 8 (𝑣𝐴 → (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (Fun 𝑣 ∧ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣))))
12 funeq 5101 . . . . . . . . . 10 (𝑧 = 𝑣 → (Fun 𝑧 ↔ Fun 𝑣))
1312biimprcd 159 . . . . . . . . 9 (Fun 𝑣 → (𝑧 = 𝑣 → Fun 𝑧))
14 sseq2 3087 . . . . . . . . . . . . . . 15 (𝑔 = 𝑥 → (𝑣𝑔𝑣𝑥))
15 sseq1 3086 . . . . . . . . . . . . . . 15 (𝑔 = 𝑥 → (𝑔𝑣𝑥𝑣))
1614, 15orbi12d 765 . . . . . . . . . . . . . 14 (𝑔 = 𝑥 → ((𝑣𝑔𝑔𝑣) ↔ (𝑣𝑥𝑥𝑣)))
1716rspcv 2756 . . . . . . . . . . . . 13 (𝑥𝐴 → (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑣𝑥𝑥𝑣)))
18 cnvss 4672 . . . . . . . . . . . . . . . 16 (𝑣𝑥𝑣𝑥)
19 cnvss 4672 . . . . . . . . . . . . . . . 16 (𝑥𝑣𝑥𝑣)
2018, 19orim12i 731 . . . . . . . . . . . . . . 15 ((𝑣𝑥𝑥𝑣) → (𝑣𝑥𝑥𝑣))
21 sseq12 3088 . . . . . . . . . . . . . . . . 17 ((𝑧 = 𝑣𝑤 = 𝑥) → (𝑧𝑤𝑣𝑥))
2221ancoms 266 . . . . . . . . . . . . . . . 16 ((𝑤 = 𝑥𝑧 = 𝑣) → (𝑧𝑤𝑣𝑥))
23 sseq12 3088 . . . . . . . . . . . . . . . 16 ((𝑤 = 𝑥𝑧 = 𝑣) → (𝑤𝑧𝑥𝑣))
2422, 23orbi12d 765 . . . . . . . . . . . . . . 15 ((𝑤 = 𝑥𝑧 = 𝑣) → ((𝑧𝑤𝑤𝑧) ↔ (𝑣𝑥𝑥𝑣)))
2520, 24syl5ibrcom 156 . . . . . . . . . . . . . 14 ((𝑣𝑥𝑥𝑣) → ((𝑤 = 𝑥𝑧 = 𝑣) → (𝑧𝑤𝑤𝑧)))
2625expd 256 . . . . . . . . . . . . 13 ((𝑣𝑥𝑥𝑣) → (𝑤 = 𝑥 → (𝑧 = 𝑣 → (𝑧𝑤𝑤𝑧))))
2717, 26syl6com 35 . . . . . . . . . . . 12 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑥𝐴 → (𝑤 = 𝑥 → (𝑧 = 𝑣 → (𝑧𝑤𝑤𝑧)))))
2827rexlimdv 2522 . . . . . . . . . . 11 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (∃𝑥𝐴 𝑤 = 𝑥 → (𝑧 = 𝑣 → (𝑧𝑤𝑤𝑧))))
2928com23 78 . . . . . . . . . 10 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑧 = 𝑣 → (∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))
3029alrimdv 1830 . . . . . . . . 9 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑧 = 𝑣 → ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))
3113, 30anim12ii 338 . . . . . . . 8 ((Fun 𝑣 ∧ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣)) → (𝑧 = 𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
3211, 31syl6com 35 . . . . . . 7 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (𝑣𝐴 → (𝑧 = 𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))))
3332rexlimdv 2522 . . . . . 6 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (∃𝑣𝐴 𝑧 = 𝑣 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
343, 33syl5bi 151 . . . . 5 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (∃𝑥𝐴 𝑧 = 𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
3534alrimiv 1828 . . . 4 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → ∀𝑧(∃𝑥𝐴 𝑧 = 𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
36 df-ral 2395 . . . . 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 2660 . . . . . . . 8 𝑧 ∈ V
38 eqeq1 2121 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 = 𝑥𝑧 = 𝑥))
3938rexbidv 2412 . . . . . . . 8 (𝑦 = 𝑧 → (∃𝑥𝐴 𝑦 = 𝑥 ↔ ∃𝑥𝐴 𝑧 = 𝑥))
4037, 39elab 2798 . . . . . . 7 (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} ↔ ∃𝑥𝐴 𝑧 = 𝑥)
41 eqeq1 2121 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦 = 𝑥𝑤 = 𝑥))
4241rexbidv 2412 . . . . . . . . 9 (𝑦 = 𝑤 → (∃𝑥𝐴 𝑦 = 𝑥 ↔ ∃𝑥𝐴 𝑤 = 𝑥))
4342ralab 2813 . . . . . . . 8 (∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧) ↔ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))
4443anbi2i 450 . . . . . . 7 ((Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧)) ↔ (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))
4540, 44imbi12i 238 . . . . . 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 1429 . . . . 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 184 . . . 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 121 . . 3 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → ∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧)))
49 fununi 5149 . . 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 4685 . . . 4 𝐴 = 𝑥𝐴 𝑥
52 vex 2660 . . . . . 6 𝑥 ∈ V
5352cnvex 5035 . . . . 5 𝑥 ∈ V
5453dfiun2 3813 . . . 4 𝑥𝐴 𝑥 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥}
5551, 54eqtri 2135 . . 3 𝐴 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥}
5655funeqi 5102 . 2 (Fun 𝐴 ↔ Fun {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥})
5750, 56sylibr 133 1 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 680  ∀wal 1312   = wceq 1314   ∈ wcel 1463  {cab 2101  ∀wral 2390  ∃wrex 2391   ⊆ wss 3037  ∪ cuni 3702  ∪ ciun 3779  ◡ccnv 4498  Fun wfun 5075 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-fun 5083 This theorem is referenced by:  fun11uni  5151
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