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Theorem isofrlem 6550
Description: Lemma for isofr 6552. (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 6533 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
31, 2syl 17 . . . . . 6 (𝜑𝐻:𝐴1-1-onto𝐵)
4 f1ofn 6100 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
5 n0 3912 . . . . . . . . . 10 (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦𝑥)
6 fnfvima 6456 . . . . . . . . . . . . 13 ((𝐻 Fn 𝐴𝑥𝐴𝑦𝑥) → (𝐻𝑦) ∈ (𝐻𝑥))
7 ne0i 3902 . . . . . . . . . . . . 13 ((𝐻𝑦) ∈ (𝐻𝑥) → (𝐻𝑥) ≠ ∅)
86, 7syl 17 . . . . . . . . . . . 12 ((𝐻 Fn 𝐴𝑥𝐴𝑦𝑥) → (𝐻𝑥) ≠ ∅)
983expia 1264 . . . . . . . . . . 11 ((𝐻 Fn 𝐴𝑥𝐴) → (𝑦𝑥 → (𝐻𝑥) ≠ ∅))
109exlimdv 1858 . . . . . . . . . 10 ((𝐻 Fn 𝐴𝑥𝐴) → (∃𝑦 𝑦𝑥 → (𝐻𝑥) ≠ ∅))
115, 10syl5bi 232 . . . . . . . . 9 ((𝐻 Fn 𝐴𝑥𝐴) → (𝑥 ≠ ∅ → (𝐻𝑥) ≠ ∅))
1211expimpd 628 . . . . . . . 8 (𝐻 Fn 𝐴 → ((𝑥𝐴𝑥 ≠ ∅) → (𝐻𝑥) ≠ ∅))
134, 12syl 17 . . . . . . 7 (𝐻:𝐴1-1-onto𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → (𝐻𝑥) ≠ ∅))
14 f1ofo 6106 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴onto𝐵)
15 imassrn 5441 . . . . . . . . 9 (𝐻𝑥) ⊆ ran 𝐻
16 forn 6080 . . . . . . . . 9 (𝐻:𝐴onto𝐵 → ran 𝐻 = 𝐵)
1715, 16syl5sseq 3637 . . . . . . . 8 (𝐻:𝐴onto𝐵 → (𝐻𝑥) ⊆ 𝐵)
1814, 17syl 17 . . . . . . 7 (𝐻:𝐴1-1-onto𝐵 → (𝐻𝑥) ⊆ 𝐵)
1913, 18jctild 565 . . . . . 6 (𝐻:𝐴1-1-onto𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → ((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅)))
203, 19syl 17 . . . . 5 (𝜑 → ((𝑥𝐴𝑥 ≠ ∅) → ((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅)))
21 dffr3 5462 . . . . . 6 (𝑆 Fr 𝐵 ↔ ∀𝑧((𝑧𝐵𝑧 ≠ ∅) → ∃𝑤𝑧 (𝑧 ∩ (𝑆 “ {𝑤})) = ∅))
22 isofrlem.2 . . . . . . 7 (𝜑 → (𝐻𝑥) ∈ V)
23 sseq1 3610 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → (𝑧𝐵 ↔ (𝐻𝑥) ⊆ 𝐵))
24 neeq1 2852 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → (𝑧 ≠ ∅ ↔ (𝐻𝑥) ≠ ∅))
2523, 24anbi12d 746 . . . . . . . . 9 (𝑧 = (𝐻𝑥) → ((𝑧𝐵𝑧 ≠ ∅) ↔ ((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅)))
26 ineq1 3790 . . . . . . . . . . 11 (𝑧 = (𝐻𝑥) → (𝑧 ∩ (𝑆 “ {𝑤})) = ((𝐻𝑥) ∩ (𝑆 “ {𝑤})))
2726eqeq1d 2623 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → ((𝑧 ∩ (𝑆 “ {𝑤})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅))
2827rexeqbi1dv 3139 . . . . . . . . 9 (𝑧 = (𝐻𝑥) → (∃𝑤𝑧 (𝑧 ∩ (𝑆 “ {𝑤})) = ∅ ↔ ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅))
2925, 28imbi12d 334 . . . . . . . 8 (𝑧 = (𝐻𝑥) → (((𝑧𝐵𝑧 ≠ ∅) → ∃𝑤𝑧 (𝑧 ∩ (𝑆 “ {𝑤})) = ∅) ↔ (((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅) → ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)))
3029spcgv 3282 . . . . . . 7 ((𝐻𝑥) ∈ V → (∀𝑧((𝑧𝐵𝑧 ≠ ∅) → ∃𝑤𝑧 (𝑧 ∩ (𝑆 “ {𝑤})) = ∅) → (((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅) → ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)))
3122, 30syl 17 . . . . . 6 (𝜑 → (∀𝑧((𝑧𝐵𝑧 ≠ ∅) → ∃𝑤𝑧 (𝑧 ∩ (𝑆 “ {𝑤})) = ∅) → (((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅) → ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)))
3221, 31syl5bi 232 . . . . 5 (𝜑 → (𝑆 Fr 𝐵 → (((𝐻𝑥) ⊆ 𝐵 ∧ (𝐻𝑥) ≠ ∅) → ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)))
3320, 32syl5d 73 . . . 4 (𝜑 → (𝑆 Fr 𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)))
343adantr 481 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐻:𝐴1-1-onto𝐵)
35 f1ofun 6101 . . . . . . . . . . 11 (𝐻:𝐴1-1-onto𝐵 → Fun 𝐻)
3634, 35syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐴) → Fun 𝐻)
37 simpl 473 . . . . . . . . . 10 ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → 𝑤 ∈ (𝐻𝑥))
38 fvelima 6210 . . . . . . . . . 10 ((Fun 𝐻𝑤 ∈ (𝐻𝑥)) → ∃𝑦𝑥 (𝐻𝑦) = 𝑤)
3936, 37, 38syl2an 494 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ (𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)) → ∃𝑦𝑥 (𝐻𝑦) = 𝑤)
40 simpr 477 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)
41 ssel 3581 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐴 → (𝑦𝑥𝑦𝐴))
4241imdistani 725 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐴𝑦𝑥) → (𝑥𝐴𝑦𝐴))
43 isomin 6547 . . . . . . . . . . . . . . . . . 18 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {(𝐻𝑦)})) = ∅))
441, 42, 43syl2an 494 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥𝐴𝑦𝑥)) → ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {(𝐻𝑦)})) = ∅))
45 sneq 4163 . . . . . . . . . . . . . . . . . . . 20 ((𝐻𝑦) = 𝑤 → {(𝐻𝑦)} = {𝑤})
4645imaeq2d 5430 . . . . . . . . . . . . . . . . . . 19 ((𝐻𝑦) = 𝑤 → (𝑆 “ {(𝐻𝑦)}) = (𝑆 “ {𝑤}))
4746ineq2d 3797 . . . . . . . . . . . . . . . . . 18 ((𝐻𝑦) = 𝑤 → ((𝐻𝑥) ∩ (𝑆 “ {(𝐻𝑦)})) = ((𝐻𝑥) ∩ (𝑆 “ {𝑤})))
4847eqeq1d 2623 . . . . . . . . . . . . . . . . 17 ((𝐻𝑦) = 𝑤 → (((𝐻𝑥) ∩ (𝑆 “ {(𝐻𝑦)})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅))
4944, 48sylan9bb 735 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐴𝑦𝑥)) ∧ (𝐻𝑦) = 𝑤) → ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅))
5040, 49syl5ibr 236 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝐴𝑦𝑥)) ∧ (𝐻𝑦) = 𝑤) → ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
5150exp42 638 . . . . . . . . . . . . . 14 (𝜑 → (𝑥𝐴 → (𝑦𝑥 → ((𝐻𝑦) = 𝑤 → ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))))
5251imp 445 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝑦𝑥 → ((𝐻𝑦) = 𝑤 → ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))))
5352com3l 89 . . . . . . . . . . . 12 (𝑦𝑥 → ((𝐻𝑦) = 𝑤 → ((𝜑𝑥𝐴) → ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))))
5453com4t 93 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → (𝑦𝑥 → ((𝐻𝑦) = 𝑤 → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))))
5554imp 445 . . . . . . . . . 10 (((𝜑𝑥𝐴) ∧ (𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)) → (𝑦𝑥 → ((𝐻𝑦) = 𝑤 → (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
5655reximdvai 3010 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ (𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)) → (∃𝑦𝑥 (𝐻𝑦) = 𝑤 → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
5739, 56mpd 15 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ (𝑤 ∈ (𝐻𝑥) ∧ ((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅)) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)
5857rexlimdvaa 3026 . . . . . . 7 ((𝜑𝑥𝐴) → (∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅ → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
5958ex 450 . . . . . 6 (𝜑 → (𝑥𝐴 → (∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅ → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
6059adantrd 484 . . . . 5 (𝜑 → ((𝑥𝐴𝑥 ≠ ∅) → (∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅ → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
6160a2d 29 . . . 4 (𝜑 → (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑤 ∈ (𝐻𝑥)((𝐻𝑥) ∩ (𝑆 “ {𝑤})) = ∅) → ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
6233, 61syld 47 . . 3 (𝜑 → (𝑆 Fr 𝐵 → ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
6362alrimdv 1854 . 2 (𝜑 → (𝑆 Fr 𝐵 → ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅)))
64 dffr3 5462 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
6563, 64syl6ibr 242 1 (𝜑 → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2908  Vcvv 3189  cin 3558  wss 3559  c0 3896  {csn 4153   Fr wfr 5035  ccnv 5078  ran crn 5080  cima 5082  Fun wfun 5846   Fn wfn 5847  ontowfo 5850  1-1-ontowf1o 5851  cfv 5852   Isom wiso 5853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-fr 5038  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861
This theorem is referenced by:  isofr  6552  isofr2  6554  isowe2  6560
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