MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isofrlem Structured version   Visualization version   GIF version

Theorem isofrlem 6735
Description: Lemma for isofr 6737. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
Hypotheses
Ref Expression
isofrlem.1 (𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
isofrlem.2 (𝜑 → (𝐻𝑥) ∈ V)
Assertion
Ref Expression
isofrlem (𝜑 → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐻   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆

Proof of Theorem isofrlem
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isofrlem.1 . . . . . . 7 (𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2 isof1o 6718 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
31, 2syl 17 . . . . . 6 (𝜑𝐻:𝐴1-1-onto𝐵)
4 f1ofn 6280 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
5 n0 4079 . . . . . . . . . 10 (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦𝑥)
6 fnfvima 6641 . . . . . . . . . . . . 13 ((𝐻 Fn 𝐴𝑥𝐴𝑦𝑥) → (𝐻𝑦) ∈ (𝐻𝑥))
76ne0d 4070 . . . . . . . . . . . 12 ((𝐻 Fn 𝐴𝑥𝐴𝑦𝑥) → (𝐻𝑥) ≠ ∅)
873expia 1114 . . . . . . . . . . 11 ((𝐻 Fn 𝐴𝑥𝐴) → (𝑦𝑥 → (𝐻𝑥) ≠ ∅))
98exlimdv 2013 . . . . . . . . . 10 ((𝐻 Fn 𝐴𝑥𝐴) → (∃𝑦 𝑦𝑥 → (𝐻𝑥) ≠ ∅))
105, 9syl5bi 232 . . . . . . . . 9 ((𝐻 Fn 𝐴𝑥𝐴) → (𝑥 ≠ ∅ → (𝐻𝑥) ≠ ∅))
1110expimpd 441 . . . . . . . 8 (𝐻 Fn 𝐴 → ((𝑥𝐴𝑥 ≠ ∅) → (𝐻𝑥) ≠ ∅))
124, 11syl 17 . . . . . . 7 (𝐻:𝐴1-1-onto𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → (𝐻𝑥) ≠ ∅))
13 f1ofo 6286 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴onto𝐵)
14 imassrn 5617 . . . . . . . . 9 (𝐻𝑥) ⊆ ran 𝐻
15 forn 6260 . . . . . . . . 9 (𝐻:𝐴onto𝐵 → ran 𝐻 = 𝐵)
1614, 15syl5sseq 3802 . . . . . . . 8 (𝐻:𝐴onto𝐵 → (𝐻𝑥) ⊆ 𝐵)
1713, 16syl 17 . . . . . . 7 (𝐻:𝐴1-1-onto𝐵 → (𝐻𝑥) ⊆ 𝐵)
1812, 17jctild 515 . . . . . 6 (𝐻:𝐴1-1-onto𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → ((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅)))
193, 18syl 17 . . . . 5 (𝜑 → ((𝑥𝐴𝑥 ≠ ∅) → ((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅)))
20 dffr3 5638 . . . . . 6 (𝑆 Fr 𝐵 ↔ ∀𝑧((𝑧𝐵𝑧 ≠ ∅) → ∃𝑤𝑧 (𝑧 ∩ (𝑆 “ {𝑤})) = ∅))
21 isofrlem.2 . . . . . . 7 (𝜑 → (𝐻𝑥) ∈ V)
22 sseq1 3775 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → (𝑧𝐵 ↔ (𝐻𝑥) ⊆ 𝐵))
23 neeq1 3005 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → (𝑧 ≠ ∅ ↔ (𝐻𝑥) ≠ ∅))
2422, 23anbi12d 616 . . . . . . . . 9 (𝑧 = (𝐻𝑥) → ((𝑧𝐵𝑧 ≠ ∅) ↔ ((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅)))
25 ineq1 3958 . . . . . . . . . . 11 (𝑧 = (𝐻𝑥) → (𝑧 ∩ (𝑆 “ {𝑤})) = ((𝐻𝑥) ∩ (𝑆 “ {𝑤})))
2625eqeq1d 2773 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → ((𝑧 ∩ (𝑆 “ {𝑤})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅))
2726rexeqbi1dv 3296 . . . . . . . . 9 (𝑧 = (𝐻𝑥) → (∃𝑤𝑧 (𝑧 ∩ (𝑆 “ {𝑤})) = ∅ ↔ ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅))
2824, 27imbi12d 333 . . . . . . . 8 (𝑧 = (𝐻𝑥) → (((𝑧𝐵𝑧 ≠ ∅) → ∃𝑤𝑧 (𝑧 ∩ (𝑆 “ {𝑤})) = ∅) ↔ (((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅) → ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)))
2928spcgv 3444 . . . . . . 7 ((𝐻𝑥) ∈ V → (∀𝑧((𝑧𝐵𝑧 ≠ ∅) → ∃𝑤𝑧 (𝑧 ∩ (𝑆 “ {𝑤})) = ∅) → (((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅) → ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)))
3021, 29syl 17 . . . . . 6 (𝜑 → (∀𝑧((𝑧𝐵𝑧 ≠ ∅) → ∃𝑤𝑧 (𝑧 ∩ (𝑆 “ {𝑤})) = ∅) → (((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅) → ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)))
3120, 30syl5bi 232 . . . . 5 (𝜑 → (𝑆 Fr 𝐵 → (((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅) → ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)))
3219, 31syl5d 73 . . . 4 (𝜑 → (𝑆 Fr 𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)))
333adantr 466 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐻:𝐴1-1-onto𝐵)
34 f1ofun 6281 . . . . . . . . . . 11 (𝐻:𝐴1-1-onto𝐵 → Fun 𝐻)
3533, 34syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐴) → Fun 𝐻)
36 simpl 468 . . . . . . . . . 10 ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → 𝑤 ∈ (𝐻𝑥))
37 fvelima 6392 . . . . . . . . . 10 ((Fun 𝐻𝑤 ∈ (𝐻𝑥)) → ∃𝑦𝑥 (𝐻𝑦) = 𝑤)
3835, 36, 37syl2an 583 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ (𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)) → ∃𝑦𝑥 (𝐻𝑦) = 𝑤)
39 simpr 471 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)
40 ssel 3746 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐴 → (𝑦𝑥𝑦𝐴))
4140imdistani 558 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐴𝑦𝑥) → (𝑥𝐴𝑦𝐴))
42 isomin 6732 . . . . . . . . . . . . . . . . . 18 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {(𝐻𝑦)})) = ∅))
431, 41, 42syl2an 583 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥𝐴𝑦𝑥)) → ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {(𝐻𝑦)})) = ∅))
44 sneq 4327 . . . . . . . . . . . . . . . . . . . 20 ((𝐻𝑦) = 𝑤 → {(𝐻𝑦)} = {𝑤})
4544imaeq2d 5606 . . . . . . . . . . . . . . . . . . 19 ((𝐻𝑦) = 𝑤 → (𝑆 “ {(𝐻𝑦)}) = (𝑆 “ {𝑤}))
4645ineq2d 3965 . . . . . . . . . . . . . . . . . 18 ((𝐻𝑦) = 𝑤 → ((𝐻𝑥) ∩ (𝑆 “ {(𝐻𝑦)})) = ((𝐻𝑥) ∩ (𝑆 “ {𝑤})))
4746eqeq1d 2773 . . . . . . . . . . . . . . . . 17 ((𝐻𝑦) = 𝑤 → (((𝐻𝑥) ∩ (𝑆 “ {(𝐻𝑦)})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅))
4843, 47sylan9bb 499 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐴𝑦𝑥)) ∧ (𝐻𝑦) = 𝑤) → ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅))
4939, 48syl5ibr 236 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝐴𝑦𝑥)) ∧ (𝐻𝑦) = 𝑤) → ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
5049exp42 422 . . . . . . . . . . . . . 14 (𝜑 → (𝑥𝐴 → (𝑦𝑥 → ((𝐻𝑦) = 𝑤 → ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))))
5150imp 393 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝑦𝑥 → ((𝐻𝑦) = 𝑤 → ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))))
5251com3l 89 . . . . . . . . . . . 12 (𝑦𝑥 → ((𝐻𝑦) = 𝑤 → ((𝜑𝑥𝐴) → ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))))
5352com4t 93 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → (𝑦𝑥 → ((𝐻𝑦) = 𝑤 → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))))
5453imp 393 . . . . . . . . . 10 (((𝜑𝑥𝐴) ∧ (𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)) → (𝑦𝑥 → ((𝐻𝑦) = 𝑤 → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
5554reximdvai 3163 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ (𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)) → (∃𝑦𝑥 (𝐻𝑦) = 𝑤 → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
5638, 55mpd 15 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ (𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)
5756rexlimdvaa 3180 . . . . . . 7 ((𝜑𝑥𝐴) → (∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅ → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
5857ex 397 . . . . . 6 (𝜑 → (𝑥𝐴 → (∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅ → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
5958adantrd 479 . . . . 5 (𝜑 → ((𝑥𝐴𝑥 ≠ ∅) → (∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅ → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
6059a2d 29 . . . 4 (𝜑 → (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
6132, 60syld 47 . . 3 (𝜑 → (𝑆 Fr 𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
6261alrimdv 2009 . 2 (𝜑 → (𝑆 Fr 𝐵 → ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
63 dffr3 5638 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
6462, 63syl6ibr 242 1 (𝜑 → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071  wal 1629   = wceq 1631  wex 1852  wcel 2145  wne 2943  wrex 3062  Vcvv 3351  cin 3722  wss 3723  c0 4063  {csn 4317   Fr wfr 5206  ccnv 5249  ran crn 5251  cima 5253  Fun wfun 6024   Fn wfn 6025  ontowfo 6028  1-1-ontowf1o 6029  cfv 6030   Isom wiso 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-fr 5209  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039
This theorem is referenced by:  isofr  6737  isofr2  6739  isowe2  6745
  Copyright terms: Public domain W3C validator