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Theorem supisoex 6481
Description: Lemma for supisoti 6482. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypotheses
Ref Expression
supiso.1 (𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
supiso.2 (𝜑𝐶𝐴)
supisoex.3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
Assertion
Ref Expression
supisoex (𝜑 → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
Distinct variable groups:   𝑣,𝑢,𝑤,𝑥,𝑦,𝑧,𝐴   𝑢,𝐶,𝑣,𝑤,𝑥,𝑦,𝑧   𝜑,𝑢,𝑤   𝑢,𝐹,𝑣,𝑤,𝑥,𝑦,𝑧   𝑢,𝑅,𝑤,𝑥,𝑦,𝑧   𝑢,𝑆,𝑣,𝑤,𝑥,𝑦,𝑧   𝑢,𝐵,𝑣,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑣)   𝑅(𝑣)

Proof of Theorem supisoex
StepHypRef Expression
1 supisoex.3 . 2 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
2 supiso.1 . . 3 (𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
3 supiso.2 . . 3 (𝜑𝐶𝐴)
4 simpl 107 . . . . . 6 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
5 simpr 108 . . . . . 6 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → 𝐶𝐴)
64, 5supisolem 6480 . . . . 5 (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐴) → ((∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
7 isof1o 5478 . . . . . . . 8 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
8 f1of 5157 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
94, 7, 83syl 17 . . . . . . 7 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → 𝐹:𝐴𝐵)
109ffvelrnda 5334 . . . . . 6 (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
11 breq1 3796 . . . . . . . . . . 11 (𝑢 = (𝐹𝑥) → (𝑢𝑆𝑤 ↔ (𝐹𝑥)𝑆𝑤))
1211notbid 625 . . . . . . . . . 10 (𝑢 = (𝐹𝑥) → (¬ 𝑢𝑆𝑤 ↔ ¬ (𝐹𝑥)𝑆𝑤))
1312ralbidv 2369 . . . . . . . . 9 (𝑢 = (𝐹𝑥) → (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ↔ ∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤))
14 breq2 3797 . . . . . . . . . . 11 (𝑢 = (𝐹𝑥) → (𝑤𝑆𝑢𝑤𝑆(𝐹𝑥)))
1514imbi1d 229 . . . . . . . . . 10 (𝑢 = (𝐹𝑥) → ((𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣) ↔ (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
1615ralbidv 2369 . . . . . . . . 9 (𝑢 = (𝐹𝑥) → (∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣) ↔ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
1713, 16anbi12d 457 . . . . . . . 8 (𝑢 = (𝐹𝑥) → ((∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
1817rspcev 2702 . . . . . . 7 (((𝐹𝑥) ∈ 𝐵 ∧ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))) → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
1918ex 113 . . . . . 6 ((𝐹𝑥) ∈ 𝐵 → ((∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)) → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
2010, 19syl 14 . . . . 5 (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐴) → ((∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)) → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
216, 20sylbid 148 . . . 4 (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐴) → ((∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)) → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
2221rexlimdva 2478 . . 3 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)) → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
232, 3, 22syl2anc 403 . 2 (𝜑 → (∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)) → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
241, 23mpd 13 1 (𝜑 → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102   = wceq 1285  wcel 1434  wral 2349  wrex 2350  wss 2974   class class class wbr 3793  cima 4374  wf 4928  1-1-ontowf1o 4931  cfv 4932   Isom wiso 4933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-isom 4941
This theorem is referenced by: (None)
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