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Theorem funcnvuni 4911
Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 4903 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
funcnvuni (f A (Fun f g A (fg gf)) → 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 4452 . . . . . . . 8 (x = vx = v)
21eqeq2d 2048 . . . . . . 7 (x = v → (z = xz = v))
32cbvrexv 2528 . . . . . 6 (x A z = xv A z = v)
4 cnveq 4452 . . . . . . . . . . 11 (f = vf = v)
54funeqd 4866 . . . . . . . . . 10 (f = v → (Fun f ↔ Fun v))
6 sseq1 2960 . . . . . . . . . . . 12 (f = v → (fgvg))
7 sseq2 2961 . . . . . . . . . . . 12 (f = v → (gfgv))
86, 7orbi12d 706 . . . . . . . . . . 11 (f = v → ((fg gf) ↔ (vg gv)))
98ralbidv 2320 . . . . . . . . . 10 (f = v → (g A (fg gf) ↔ g A (vg gv)))
105, 9anbi12d 442 . . . . . . . . 9 (f = v → ((Fun f g A (fg gf)) ↔ (Fun v g A (vg gv))))
1110rspcv 2646 . . . . . . . 8 (v A → (f A (Fun f g A (fg gf)) → (Fun v g A (vg gv))))
12 funeq 4864 . . . . . . . . . 10 (z = v → (Fun z ↔ Fun v))
1312biimprcd 149 . . . . . . . . 9 (Fun v → (z = v → Fun z))
14 sseq2 2961 . . . . . . . . . . . . . . 15 (g = x → (vgvx))
15 sseq1 2960 . . . . . . . . . . . . . . 15 (g = x → (gvxv))
1614, 15orbi12d 706 . . . . . . . . . . . . . 14 (g = x → ((vg gv) ↔ (vx xv)))
1716rspcv 2646 . . . . . . . . . . . . 13 (x A → (g A (vg gv) → (vx xv)))
18 cnvss 4451 . . . . . . . . . . . . . . . 16 (vxvx)
19 cnvss 4451 . . . . . . . . . . . . . . . 16 (xvxv)
2018, 19orim12i 675 . . . . . . . . . . . . . . 15 ((vx xv) → (vx xv))
21 sseq12 2962 . . . . . . . . . . . . . . . . 17 ((z = v w = x) → (zwvx))
2221ancoms 255 . . . . . . . . . . . . . . . 16 ((w = x z = v) → (zwvx))
23 sseq12 2962 . . . . . . . . . . . . . . . 16 ((w = x z = v) → (wzxv))
2422, 23orbi12d 706 . . . . . . . . . . . . . . 15 ((w = x z = v) → ((zw wz) ↔ (vx xv)))
2520, 24syl5ibrcom 146 . . . . . . . . . . . . . 14 ((vx xv) → ((w = x z = v) → (zw wz)))
2625expd 245 . . . . . . . . . . . . 13 ((vx xv) → (w = x → (z = v → (zw wz))))
2717, 26syl6com 31 . . . . . . . . . . . 12 (g A (vg gv) → (x A → (w = x → (z = v → (zw wz)))))
2827rexlimdv 2426 . . . . . . . . . . 11 (g A (vg gv) → (x A w = x → (z = v → (zw wz))))
2928com23 72 . . . . . . . . . 10 (g A (vg gv) → (z = v → (x A w = x → (zw wz))))
3029alrimdv 1753 . . . . . . . . 9 (g A (vg gv) → (z = vw(x A w = x → (zw wz))))
3113, 30anim12ii 325 . . . . . . . 8 ((Fun v g A (vg gv)) → (z = v → (Fun z w(x A w = x → (zw wz)))))
3211, 31syl6com 31 . . . . . . 7 (f A (Fun f g A (fg gf)) → (v A → (z = v → (Fun z w(x A w = x → (zw wz))))))
3332rexlimdv 2426 . . . . . 6 (f A (Fun f g A (fg gf)) → (v A z = v → (Fun z w(x A w = x → (zw wz)))))
343, 33syl5bi 141 . . . . 5 (f A (Fun f g A (fg gf)) → (x A z = x → (Fun z w(x A w = x → (zw wz)))))
3534alrimiv 1751 . . . 4 (f A (Fun f g A (fg gf)) → z(x A z = x → (Fun z w(x A w = x → (zw wz)))))
36 df-ral 2305 . . . . 5 (z {yx A y = x} (Fun z w {yx A y = x} (zw wz)) ↔ z(z {yx A y = x} → (Fun z w {yx A y = x} (zw wz))))
37 vex 2554 . . . . . . . 8 z V
38 eqeq1 2043 . . . . . . . . 9 (y = z → (y = xz = x))
3938rexbidv 2321 . . . . . . . 8 (y = z → (x A y = xx A z = x))
4037, 39elab 2681 . . . . . . 7 (z {yx A y = x} ↔ x A z = x)
41 eqeq1 2043 . . . . . . . . . 10 (y = w → (y = xw = x))
4241rexbidv 2321 . . . . . . . . 9 (y = w → (x A y = xx A w = x))
4342ralab 2695 . . . . . . . 8 (w {yx A y = x} (zw wz) ↔ w(x A w = x → (zw wz)))
4443anbi2i 430 . . . . . . 7 ((Fun z w {yx A y = x} (zw wz)) ↔ (Fun z w(x A w = x → (zw wz))))
4540, 44imbi12i 228 . . . . . 6 ((z {yx A y = x} → (Fun z w {yx A y = x} (zw wz))) ↔ (x A z = x → (Fun z w(x A w = x → (zw wz)))))
4645albii 1356 . . . . 5 (z(z {yx A y = x} → (Fun z w {yx A y = x} (zw wz))) ↔ z(x A z = x → (Fun z w(x A w = x → (zw wz)))))
4736, 46bitr2i 174 . . . 4 (z(x A z = x → (Fun z w(x A w = x → (zw wz)))) ↔ z {yx A y = x} (Fun z w {yx A y = x} (zw wz)))
4835, 47sylib 127 . . 3 (f A (Fun f g A (fg gf)) → z {yx A y = x} (Fun z w {yx A y = x} (zw wz)))
49 fununi 4910 . . 3 (z {yx A y = x} (Fun z w {yx A y = x} (zw wz)) → Fun {yx A y = x})
5048, 49syl 14 . 2 (f A (Fun f g A (fg gf)) → Fun {yx A y = x})
51 cnvuni 4464 . . . 4 A = x A x
52 vex 2554 . . . . . 6 x V
5352cnvex 4799 . . . . 5 x V
5453dfiun2 3682 . . . 4 x A x = {yx A y = x}
5551, 54eqtri 2057 . . 3 A = {yx A y = x}
5655funeqi 4865 . 2 (Fun A ↔ Fun {yx A y = x})
5750, 56sylibr 137 1 (f A (Fun f g A (fg gf)) → Fun A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628  wal 1240   = wceq 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301  wss 2911   cuni 3571   ciun 3648  ccnv 4287  Fun wfun 4839
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847
This theorem is referenced by:  fun11uni  4912
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