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Theorem phialllem1 4616
Description: Lemma for phiall 4618. Any set of numbers without zero is the Phi of a set. (Contributed by Scott Fenton, 14-Apr-2021.)
Hypothesis
Ref Expression
phiall.1 A V
Assertion
Ref Expression
phialllem1 ((A Nn ¬ 0c A) → x A = Phi x)
Distinct variable group:   x,A

Proof of Theorem phialllem1
Dummy variables y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2413 . . . . . . . . . . . . 13 (z = 0c → (z A ↔ 0c A))
21biimpcd 215 . . . . . . . . . . . 12 (z A → (z = 0c → 0c A))
32con3d 125 . . . . . . . . . . 11 (z A → (¬ 0c A → ¬ z = 0c))
43impcom 419 . . . . . . . . . 10 ((¬ 0c A z A) → ¬ z = 0c)
54adantll 694 . . . . . . . . 9 (((A Nn ¬ 0c A) z A) → ¬ z = 0c)
6 ssel2 3268 . . . . . . . . . . 11 ((A Nn z A) → z Nn )
76adantlr 695 . . . . . . . . . 10 (((A Nn ¬ 0c A) z A) → z Nn )
8 nnc0suc 4412 . . . . . . . . . 10 (z Nn ↔ (z = 0c x Nn z = (x +c 1c)))
97, 8sylib 188 . . . . . . . . 9 (((A Nn ¬ 0c A) z A) → (z = 0c x Nn z = (x +c 1c)))
10 orel1 371 . . . . . . . . 9 z = 0c → ((z = 0c x Nn z = (x +c 1c)) → x Nn z = (x +c 1c)))
115, 9, 10sylc 56 . . . . . . . 8 (((A Nn ¬ 0c A) z A) → x Nn z = (x +c 1c))
12 anidm 625 . . . . . . . . . 10 ((z = (x +c 1c) z = (x +c 1c)) ↔ z = (x +c 1c))
13 eqeq1 2359 . . . . . . . . . . 11 (z = w → (z = (x +c 1c) ↔ w = (x +c 1c)))
1413anbi2d 684 . . . . . . . . . 10 (z = w → ((z = (x +c 1c) z = (x +c 1c)) ↔ (z = (x +c 1c) w = (x +c 1c))))
1512, 14syl5bbr 250 . . . . . . . . 9 (z = w → (z = (x +c 1c) ↔ (z = (x +c 1c) w = (x +c 1c))))
1615rexbidv 2635 . . . . . . . 8 (z = w → (x Nn z = (x +c 1c) ↔ x Nn (z = (x +c 1c) w = (x +c 1c))))
1711, 16syl5ibcom 211 . . . . . . 7 (((A Nn ¬ 0c A) z A) → (z = wx Nn (z = (x +c 1c) w = (x +c 1c))))
18 eqtr3 2372 . . . . . . . 8 ((z = (x +c 1c) w = (x +c 1c)) → z = w)
1918rexlimivw 2734 . . . . . . 7 (x Nn (z = (x +c 1c) w = (x +c 1c)) → z = w)
2017, 19impbid1 194 . . . . . 6 (((A Nn ¬ 0c A) z A) → (z = wx Nn (z = (x +c 1c) w = (x +c 1c))))
2120rexbidva 2631 . . . . 5 ((A Nn ¬ 0c A) → (z A z = wz A x Nn (z = (x +c 1c) w = (x +c 1c))))
22 risset 2661 . . . . 5 (w Az A z = w)
23 rexcom 2772 . . . . 5 (x Nn z A (z = (x +c 1c) w = (x +c 1c)) ↔ z A x Nn (z = (x +c 1c) w = (x +c 1c)))
2421, 22, 233bitr4g 279 . . . 4 ((A Nn ¬ 0c A) → (w Ax Nn z A (z = (x +c 1c) w = (x +c 1c))))
2524abbi2dv 2468 . . 3 ((A Nn ¬ 0c A) → A = {w x Nn z A (z = (x +c 1c) w = (x +c 1c))})
26 df-phi 4565 . . . 4 Phi {y Nn z A z = (y +c 1c)} = {w x {y Nn z A z = (y +c 1c)}w = if(x Nn , (x +c 1c), x)}
27 addceq1 4383 . . . . . . . . 9 (y = x → (y +c 1c) = (x +c 1c))
2827eqeq2d 2364 . . . . . . . 8 (y = x → (z = (y +c 1c) ↔ z = (x +c 1c)))
2928rexbidv 2635 . . . . . . 7 (y = x → (z A z = (y +c 1c) ↔ z A z = (x +c 1c)))
3029rexrab 3000 . . . . . 6 (x {y Nn z A z = (y +c 1c)}w = if(x Nn , (x +c 1c), x) ↔ x Nn (z A z = (x +c 1c) w = if(x Nn , (x +c 1c), x)))
31 iftrue 3668 . . . . . . . . . 10 (x Nn → if(x Nn , (x +c 1c), x) = (x +c 1c))
3231eqeq2d 2364 . . . . . . . . 9 (x Nn → (w = if(x Nn , (x +c 1c), x) ↔ w = (x +c 1c)))
3332anbi2d 684 . . . . . . . 8 (x Nn → ((z A z = (x +c 1c) w = if(x Nn , (x +c 1c), x)) ↔ (z A z = (x +c 1c) w = (x +c 1c))))
34 r19.41v 2764 . . . . . . . 8 (z A (z = (x +c 1c) w = (x +c 1c)) ↔ (z A z = (x +c 1c) w = (x +c 1c)))
3533, 34syl6bbr 254 . . . . . . 7 (x Nn → ((z A z = (x +c 1c) w = if(x Nn , (x +c 1c), x)) ↔ z A (z = (x +c 1c) w = (x +c 1c))))
3635rexbiia 2647 . . . . . 6 (x Nn (z A z = (x +c 1c) w = if(x Nn , (x +c 1c), x)) ↔ x Nn z A (z = (x +c 1c) w = (x +c 1c)))
3730, 36bitri 240 . . . . 5 (x {y Nn z A z = (y +c 1c)}w = if(x Nn , (x +c 1c), x) ↔ x Nn z A (z = (x +c 1c) w = (x +c 1c)))
3837abbii 2465 . . . 4 {w x {y Nn z A z = (y +c 1c)}w = if(x Nn , (x +c 1c), x)} = {w x Nn z A (z = (x +c 1c) w = (x +c 1c))}
3926, 38eqtri 2373 . . 3 Phi {y Nn z A z = (y +c 1c)} = {w x Nn z A (z = (x +c 1c) w = (x +c 1c))}
4025, 39syl6eqr 2403 . 2 ((A Nn ¬ 0c A) → A = Phi {y Nn z A z = (y +c 1c)})
41 dfrab2 3530 . . . 4 {y Nn z A z = (y +c 1c)} = ({y z A z = (y +c 1c)} ∩ Nn )
42 vex 2862 . . . . . . . . 9 y V
4342elimak 4259 . . . . . . . 8 (y (kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) “k A) ↔ z Az, y kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c))
44 vex 2862 . . . . . . . . . . 11 z V
4542, 44opkelimagek 4272 . . . . . . . . . 10 (⟪y, z Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ↔ z = ((( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) “k y))
4644, 42opkelcnvk 4250 . . . . . . . . . 10 (⟪z, y kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ↔ ⟪y, z Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c))
47 dfaddc2 4381 . . . . . . . . . . 11 (y +c 1c) = ((( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) “k y)
4847eqeq2i 2363 . . . . . . . . . 10 (z = (y +c 1c) ↔ z = ((( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) “k y))
4945, 46, 483bitr4i 268 . . . . . . . . 9 (⟪z, y kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ↔ z = (y +c 1c))
5049rexbii 2639 . . . . . . . 8 (z Az, y kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ↔ z A z = (y +c 1c))
5143, 50bitri 240 . . . . . . 7 (y (kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) “k A) ↔ z A z = (y +c 1c))
5251abbi2i 2464 . . . . . 6 (kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) “k A) = {y z A z = (y +c 1c)}
53 addcexlem 4382 . . . . . . . . . 10 ( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) V
54 1cex 4142 . . . . . . . . . . . 12 1c V
5554pw1ex 4303 . . . . . . . . . . 11 11c V
5655pw1ex 4303 . . . . . . . . . 10 111c V
5753, 56imakex 4300 . . . . . . . . 9 (( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) V
5857imagekex 4312 . . . . . . . 8 Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) V
5958cnvkex 4287 . . . . . . 7 kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) V
60 phiall.1 . . . . . . 7 A V
6159, 60imakex 4300 . . . . . 6 (kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) “k A) V
6252, 61eqeltrri 2424 . . . . 5 {y z A z = (y +c 1c)} V
63 nncex 4396 . . . . 5 Nn V
6462, 63inex 4105 . . . 4 ({y z A z = (y +c 1c)} ∩ Nn ) V
6541, 64eqeltri 2423 . . 3 {y Nn z A z = (y +c 1c)} V
66 phieq 4570 . . . 4 (x = {y Nn z A z = (y +c 1c)} → Phi x = Phi {y Nn z A z = (y +c 1c)})
6766eqeq2d 2364 . . 3 (x = {y Nn z A z = (y +c 1c)} → (A = Phi xA = Phi {y Nn z A z = (y +c 1c)}))
6865, 67spcev 2946 . 2 (A = Phi {y Nn z A z = (y +c 1c)} → x A = Phi x)
6940, 68syl 15 1 ((A Nn ¬ 0c A) → x A = Phi x)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   wo 357   wa 358  wex 1541   = wceq 1642   wcel 1710  {cab 2339  wrex 2615  {crab 2618  Vcvv 2859  ccompl 3205   cdif 3206  cun 3207  cin 3208  csymdif 3209   wss 3257   ifcif 3662  copk 4057  1cc1c 4134  1cpw1 4135  kccnvk 4175   Ins2k cins2k 4176   Ins3k cins3k 4177  k cimak 4179   SIk csik 4181  Imagekcimagek 4182   Sk cssetk 4183   Nn cnnc 4373  0cc0c 4374   +c cplc 4375   Phi cphi 4562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-0c 4377  df-addc 4378  df-nnc 4379  df-phi 4565
This theorem is referenced by:  phialllem2  4617
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