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Theorem funcnvuni 7066
Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5916 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 5256 . . . . . . . 8 (𝑥 = 𝑣𝑥 = 𝑣)
21eqeq2d 2631 . . . . . . 7 (𝑥 = 𝑣 → (𝑧 = 𝑥𝑧 = 𝑣))
32cbvrexv 3160 . . . . . 6 (∃𝑥𝐴 𝑧 = 𝑥 ↔ ∃𝑣𝐴 𝑧 = 𝑣)
4 cnveq 5256 . . . . . . . . . . 11 (𝑓 = 𝑣𝑓 = 𝑣)
54funeqd 5869 . . . . . . . . . 10 (𝑓 = 𝑣 → (Fun 𝑓 ↔ Fun 𝑣))
6 sseq1 3605 . . . . . . . . . . . 12 (𝑓 = 𝑣 → (𝑓𝑔𝑣𝑔))
7 sseq2 3606 . . . . . . . . . . . 12 (𝑓 = 𝑣 → (𝑔𝑓𝑔𝑣))
86, 7orbi12d 745 . . . . . . . . . . 11 (𝑓 = 𝑣 → ((𝑓𝑔𝑔𝑓) ↔ (𝑣𝑔𝑔𝑣)))
98ralbidv 2980 . . . . . . . . . 10 (𝑓 = 𝑣 → (∀𝑔𝐴 (𝑓𝑔𝑔𝑓) ↔ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣)))
105, 9anbi12d 746 . . . . . . . . 9 (𝑓 = 𝑣 → ((Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) ↔ (Fun 𝑣 ∧ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣))))
1110rspcv 3291 . . . . . . . 8 (𝑣𝐴 → (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (Fun 𝑣 ∧ ∀𝑔𝐴 (𝑣𝑔𝑔𝑣))))
12 funeq 5867 . . . . . . . . . 10 (𝑧 = 𝑣 → (Fun 𝑧 ↔ Fun 𝑣))
1312biimprcd 240 . . . . . . . . 9 (Fun 𝑣 → (𝑧 = 𝑣 → Fun 𝑧))
14 sseq2 3606 . . . . . . . . . . . . . . 15 (𝑔 = 𝑥 → (𝑣𝑔𝑣𝑥))
15 sseq1 3605 . . . . . . . . . . . . . . 15 (𝑔 = 𝑥 → (𝑔𝑣𝑥𝑣))
1614, 15orbi12d 745 . . . . . . . . . . . . . 14 (𝑔 = 𝑥 → ((𝑣𝑔𝑔𝑣) ↔ (𝑣𝑥𝑥𝑣)))
1716rspcv 3291 . . . . . . . . . . . . 13 (𝑥𝐴 → (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑣𝑥𝑥𝑣)))
18 cnvss 5254 . . . . . . . . . . . . . . . 16 (𝑣𝑥𝑣𝑥)
19 cnvss 5254 . . . . . . . . . . . . . . . 16 (𝑥𝑣𝑥𝑣)
2018, 19orim12i 538 . . . . . . . . . . . . . . 15 ((𝑣𝑥𝑥𝑣) → (𝑣𝑥𝑥𝑣))
21 sseq12 3607 . . . . . . . . . . . . . . . . 17 ((𝑧 = 𝑣𝑤 = 𝑥) → (𝑧𝑤𝑣𝑥))
2221ancoms 469 . . . . . . . . . . . . . . . 16 ((𝑤 = 𝑥𝑧 = 𝑣) → (𝑧𝑤𝑣𝑥))
23 sseq12 3607 . . . . . . . . . . . . . . . 16 ((𝑤 = 𝑥𝑧 = 𝑣) → (𝑤𝑧𝑥𝑣))
2422, 23orbi12d 745 . . . . . . . . . . . . . . 15 ((𝑤 = 𝑥𝑧 = 𝑣) → ((𝑧𝑤𝑤𝑧) ↔ (𝑣𝑥𝑥𝑣)))
2520, 24syl5ibrcom 237 . . . . . . . . . . . . . 14 ((𝑣𝑥𝑥𝑣) → ((𝑤 = 𝑥𝑧 = 𝑣) → (𝑧𝑤𝑤𝑧)))
2625expd 452 . . . . . . . . . . . . 13 ((𝑣𝑥𝑥𝑣) → (𝑤 = 𝑥 → (𝑧 = 𝑣 → (𝑧𝑤𝑤𝑧))))
2717, 26syl6com 37 . . . . . . . . . . . 12 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑥𝐴 → (𝑤 = 𝑥 → (𝑧 = 𝑣 → (𝑧𝑤𝑤𝑧)))))
2827rexlimdv 3023 . . . . . . . . . . 11 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (∃𝑥𝐴 𝑤 = 𝑥 → (𝑧 = 𝑣 → (𝑧𝑤𝑤𝑧))))
2928com23 86 . . . . . . . . . 10 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑧 = 𝑣 → (∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))
3029alrimdv 1854 . . . . . . . . 9 (∀𝑔𝐴 (𝑣𝑔𝑔𝑣) → (𝑧 = 𝑣 → ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))
3113, 30anim12ii 593 . . . . . . . 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 3023 . . . . . 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 1852 . . . 4 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → ∀𝑧(∃𝑥𝐴 𝑧 = 𝑥 → (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))))
36 df-ral 2912 . . . . 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 3189 . . . . . . . 8 𝑧 ∈ V
38 eqeq1 2625 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 = 𝑥𝑧 = 𝑥))
3938rexbidv 3045 . . . . . . . 8 (𝑦 = 𝑧 → (∃𝑥𝐴 𝑦 = 𝑥 ↔ ∃𝑥𝐴 𝑧 = 𝑥))
4037, 39elab 3333 . . . . . . 7 (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} ↔ ∃𝑥𝐴 𝑧 = 𝑥)
41 eqeq1 2625 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦 = 𝑥𝑤 = 𝑥))
4241rexbidv 3045 . . . . . . . . 9 (𝑦 = 𝑤 → (∃𝑥𝐴 𝑦 = 𝑥 ↔ ∃𝑥𝐴 𝑤 = 𝑥))
4342ralab 3349 . . . . . . . 8 (∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧) ↔ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧)))
4443anbi2i 729 . . . . . . 7 ((Fun 𝑧 ∧ ∀𝑤 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥} (𝑧𝑤𝑤𝑧)) ↔ (Fun 𝑧 ∧ ∀𝑤(∃𝑥𝐴 𝑤 = 𝑥 → (𝑧𝑤𝑤𝑧))))
4540, 44imbi12i 340 . . . . . 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 1744 . . . . 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 5922 . . 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 5269 . . . 4 𝐴 = 𝑥𝐴 𝑥
52 vex 3189 . . . . . 6 𝑥 ∈ V
5352cnvex 7060 . . . . 5 𝑥 ∈ V
5453dfiun2 4520 . . . 4 𝑥𝐴 𝑥 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥}
5551, 54eqtri 2643 . . 3 𝐴 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝑥}
5655funeqi 5868 . 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 383  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  wss 3555   cuni 4402   ciun 4485  ccnv 5073  Fun wfun 5841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-fun 5849
This theorem is referenced by:  fun11uni  7067
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