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Theorem funcnvuni 4890
Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 4882 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 4432 . . . . . . . 8 (x = vx = v)
21eqeq2d 2029 . . . . . . 7 (x = v → (z = xz = v))
32cbvrexv 2508 . . . . . 6 (x A z = xv A z = v)
4 cnveq 4432 . . . . . . . . . . 11 (f = vf = v)
54funeqd 4845 . . . . . . . . . 10 (f = v → (Fun f ↔ Fun v))
6 sseq1 2939 . . . . . . . . . . . 12 (f = v → (fgvg))
7 sseq2 2940 . . . . . . . . . . . 12 (f = v → (gfgv))
86, 7orbi12d 694 . . . . . . . . . . 11 (f = v → ((fg gf) ↔ (vg gv)))
98ralbidv 2300 . . . . . . . . . 10 (f = v → (g A (fg gf) ↔ g A (vg gv)))
105, 9anbi12d 445 . . . . . . . . 9 (f = v → ((Fun f g A (fg gf)) ↔ (Fun v g A (vg gv))))
1110rspcv 2625 . . . . . . . 8 (v A → (f A (Fun f g A (fg gf)) → (Fun v g A (vg gv))))
12 funeq 4843 . . . . . . . . . 10 (z = v → (Fun z ↔ Fun v))
1312biimprcd 149 . . . . . . . . 9 (Fun v → (z = v → Fun z))
14 sseq2 2940 . . . . . . . . . . . . . . 15 (g = x → (vgvx))
15 sseq1 2939 . . . . . . . . . . . . . . 15 (g = x → (gvxv))
1614, 15orbi12d 694 . . . . . . . . . . . . . 14 (g = x → ((vg gv) ↔ (vx xv)))
1716rspcv 2625 . . . . . . . . . . . . 13 (x A → (g A (vg gv) → (vx xv)))
18 cnvss 4431 . . . . . . . . . . . . . . . 16 (vxvx)
19 cnvss 4431 . . . . . . . . . . . . . . . 16 (xvxv)
2018, 19orim12i 663 . . . . . . . . . . . . . . 15 ((vx xv) → (vx xv))
21 sseq12 2941 . . . . . . . . . . . . . . . . 17 ((z = v w = x) → (zwvx))
2221ancoms 255 . . . . . . . . . . . . . . . 16 ((w = x z = v) → (zwvx))
23 sseq12 2941 . . . . . . . . . . . . . . . 16 ((w = x z = v) → (wzxv))
2422, 23orbi12d 694 . . . . . . . . . . . . . . 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 2406 . . . . . . . . . . 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 1734 . . . . . . . . 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 2406 . . . . . 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 1732 . . . 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 2285 . . . . 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 2534 . . . . . . . 8 z V
38 eqeq1 2024 . . . . . . . . 9 (y = z → (y = xz = x))
3938rexbidv 2301 . . . . . . . 8 (y = z → (x A y = xx A z = x))
4037, 39elab 2660 . . . . . . 7 (z {yx A y = x} ↔ x A z = x)
41 eqeq1 2024 . . . . . . . . . 10 (y = w → (y = xw = x))
4241rexbidv 2301 . . . . . . . . 9 (y = w → (x A y = xx A w = x))
4342ralab 2674 . . . . . . . 8 (w {yx A y = x} (zw wz) ↔ w(x A w = x → (zw wz)))
4443anbi2i 433 . . . . . . 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 1335 . . . . 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 4889 . . 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 4444 . . . 4 A = x A x
52 vex 2534 . . . . . 6 x V
5352cnvex 4779 . . . . 5 x V
5453dfiun2 3661 . . . 4 x A x = {yx A y = x}
5551, 54eqtri 2038 . . 3 A = {yx A y = x}
5655funeqi 4844 . 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 616  wal 1224   = wceq 1226   wcel 1370  {cab 2004  wral 2280  wrex 2281  wss 2890   cuni 3550   ciun 3627  ccnv 4267  Fun wfun 4819
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-fun 4827
This theorem is referenced by:  fun11uni  4891
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