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Theorem isofrlem 7347
Description: Lemma for isofr 7349. (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 7330 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
31, 2syl 17 . . . . . 6 (𝜑𝐻:𝐴1-1-onto𝐵)
4 f1ofn 6839 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
5 n0 4346 . . . . . . . . . 10 (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦𝑥)
6 fnfvima 7245 . . . . . . . . . . . . 13 ((𝐻 Fn 𝐴𝑥𝐴𝑦𝑥) → (𝐻𝑦) ∈ (𝐻𝑥))
76ne0d 4335 . . . . . . . . . . . 12 ((𝐻 Fn 𝐴𝑥𝐴𝑦𝑥) → (𝐻𝑥) ≠ ∅)
873expia 1118 . . . . . . . . . . 11 ((𝐻 Fn 𝐴𝑥𝐴) → (𝑦𝑥 → (𝐻𝑥) ≠ ∅))
98exlimdv 1928 . . . . . . . . . 10 ((𝐻 Fn 𝐴𝑥𝐴) → (∃𝑦 𝑦𝑥 → (𝐻𝑥) ≠ ∅))
105, 9biimtrid 241 . . . . . . . . 9 ((𝐻 Fn 𝐴𝑥𝐴) → (𝑥 ≠ ∅ → (𝐻𝑥) ≠ ∅))
1110expimpd 452 . . . . . . . 8 (𝐻 Fn 𝐴 → ((𝑥𝐴𝑥 ≠ ∅) → (𝐻𝑥) ≠ ∅))
124, 11syl 17 . . . . . . 7 (𝐻:𝐴1-1-onto𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → (𝐻𝑥) ≠ ∅))
13 f1ofo 6845 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴onto𝐵)
14 imassrn 6075 . . . . . . . . 9 (𝐻𝑥) ⊆ ran 𝐻
15 forn 6813 . . . . . . . . 9 (𝐻:𝐴onto𝐵 → ran 𝐻 = 𝐵)
1614, 15sseqtrid 4029 . . . . . . . 8 (𝐻:𝐴onto𝐵 → (𝐻𝑥) ⊆ 𝐵)
1713, 16syl 17 . . . . . . 7 (𝐻:𝐴1-1-onto𝐵 → (𝐻𝑥) ⊆ 𝐵)
1812, 17jctild 524 . . . . . 6 (𝐻:𝐴1-1-onto𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → ((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅)))
193, 18syl 17 . . . . 5 (𝜑 → ((𝑥𝐴𝑥 ≠ ∅) → ((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅)))
20 dffr3 6104 . . . . . 6 (𝑆 Fr 𝐵 ↔ ∀𝑧((𝑧𝐵𝑧 ≠ ∅) → ∃𝑤𝑧 (𝑧 ∩ (𝑆 “ {𝑤})) = ∅))
21 isofrlem.2 . . . . . . 7 (𝜑 → (𝐻𝑥) ∈ V)
22 sseq1 4002 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → (𝑧𝐵 ↔ (𝐻𝑥) ⊆ 𝐵))
23 neeq1 2992 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → (𝑧 ≠ ∅ ↔ (𝐻𝑥) ≠ ∅))
2422, 23anbi12d 630 . . . . . . . . 9 (𝑧 = (𝐻𝑥) → ((𝑧𝐵𝑧 ≠ ∅) ↔ ((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅)))
25 ineq1 4203 . . . . . . . . . . 11 (𝑧 = (𝐻𝑥) → (𝑧 ∩ (𝑆 “ {𝑤})) = ((𝐻𝑥) ∩ (𝑆 “ {𝑤})))
2625eqeq1d 2727 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → ((𝑧 ∩ (𝑆 “ {𝑤})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅))
2726rexeqbi1dv 3323 . . . . . . . . 9 (𝑧 = (𝐻𝑥) → (∃𝑤𝑧 (𝑧 ∩ (𝑆 “ {𝑤})) = ∅ ↔ ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅))
2824, 27imbi12d 343 . . . . . . . 8 (𝑧 = (𝐻𝑥) → (((𝑧𝐵𝑧 ≠ ∅) → ∃𝑤𝑧 (𝑧 ∩ (𝑆 “ {𝑤})) = ∅) ↔ (((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅) → ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)))
2928spcgv 3580 . . . . . . 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, 30biimtrid 241 . . . . 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 479 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐻:𝐴1-1-onto𝐵)
34 f1ofun 6840 . . . . . . . . . . 11 (𝐻:𝐴1-1-onto𝐵 → Fun 𝐻)
3533, 34syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐴) → Fun 𝐻)
36 simpl 481 . . . . . . . . . 10 ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → 𝑤 ∈ (𝐻𝑥))
37 fvelima 6963 . . . . . . . . . 10 ((Fun 𝐻𝑤 ∈ (𝐻𝑥)) → ∃𝑦𝑥 (𝐻𝑦) = 𝑤)
3835, 36, 37syl2an 594 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ (𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)) → ∃𝑦𝑥 (𝐻𝑦) = 𝑤)
39 simpr 483 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)
40 ssel 3970 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐴 → (𝑦𝑥𝑦𝐴))
4140imdistani 567 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐴𝑦𝑥) → (𝑥𝐴𝑦𝐴))
42 isomin 7344 . . . . . . . . . . . . . . . . . 18 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {(𝐻𝑦)})) = ∅))
431, 41, 42syl2an 594 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥𝐴𝑦𝑥)) → ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {(𝐻𝑦)})) = ∅))
44 sneq 4640 . . . . . . . . . . . . . . . . . . . 20 ((𝐻𝑦) = 𝑤 → {(𝐻𝑦)} = {𝑤})
4544imaeq2d 6064 . . . . . . . . . . . . . . . . . . 19 ((𝐻𝑦) = 𝑤 → (𝑆 “ {(𝐻𝑦)}) = (𝑆 “ {𝑤}))
4645ineq2d 4210 . . . . . . . . . . . . . . . . . 18 ((𝐻𝑦) = 𝑤 → ((𝐻𝑥) ∩ (𝑆 “ {(𝐻𝑦)})) = ((𝐻𝑥) ∩ (𝑆 “ {𝑤})))
4746eqeq1d 2727 . . . . . . . . . . . . . . . . 17 ((𝐻𝑦) = 𝑤 → (((𝐻𝑥) ∩ (𝑆 “ {(𝐻𝑦)})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅))
4843, 47sylan9bb 508 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐴𝑦𝑥)) ∧ (𝐻𝑦) = 𝑤) → ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅))
4939, 48imbitrrid 245 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝐴𝑦𝑥)) ∧ (𝐻𝑦) = 𝑤) → ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
5049exp42 434 . . . . . . . . . . . . . 14 (𝜑 → (𝑥𝐴 → (𝑦𝑥 → ((𝐻𝑦) = 𝑤 → ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))))
5150imp 405 . . . . . . . . . . . . 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 405 . . . . . . . . . 10 (((𝜑𝑥𝐴) ∧ (𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)) → (𝑦𝑥 → ((𝐻𝑦) = 𝑤 → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
5554reximdvai 3154 . . . . . . . . 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 3145 . . . . . . 7 ((𝜑𝑥𝐴) → (∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅ → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
5857ex 411 . . . . . 6 (𝜑 → (𝑥𝐴 → (∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅ → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
5958adantrd 490 . . . . 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 1924 . 2 (𝜑 → (𝑆 Fr 𝐵 → ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
63 dffr3 6104 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
6462, 63imbitrrdi 251 1 (𝜑 → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084  wal 1531   = wceq 1533  wex 1773  wcel 2098  wne 2929  wrex 3059  Vcvv 3461  cin 3943  wss 3944  c0 4322  {csn 4630   Fr wfr 5630  ccnv 5677  ran crn 5679  cima 5681  Fun wfun 6543   Fn wfn 6544  ontowfo 6547  1-1-ontowf1o 6548  cfv 6549   Isom wiso 6550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-fr 5633  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558
This theorem is referenced by:  isofr  7349  isofr2  7351  isowe2  7357
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