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Theorem supisoex 6702
Description: Lemma for supisoti 6703. (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 6701 . . . . 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 5586 . . . . . . . 8 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
8 f1of 5253 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
94, 7, 83syl 17 . . . . . . 7 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → 𝐹:𝐴𝐵)
109ffvelrnda 5434 . . . . . 6 (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
11 breq1 3848 . . . . . . . . . . 11 (𝑢 = (𝐹𝑥) → (𝑢𝑆𝑤 ↔ (𝐹𝑥)𝑆𝑤))
1211notbid 627 . . . . . . . . . 10 (𝑢 = (𝐹𝑥) → (¬ 𝑢𝑆𝑤 ↔ ¬ (𝐹𝑥)𝑆𝑤))
1312ralbidv 2380 . . . . . . . . 9 (𝑢 = (𝐹𝑥) → (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ↔ ∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤))
14 breq2 3849 . . . . . . . . . . 11 (𝑢 = (𝐹𝑥) → (𝑤𝑆𝑢𝑤𝑆(𝐹𝑥)))
1514imbi1d 229 . . . . . . . . . 10 (𝑢 = (𝐹𝑥) → ((𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣) ↔ (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
1615ralbidv 2380 . . . . . . . . 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 2722 . . . . . . 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 2489 . . 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 1289  wcel 1438  wral 2359  wrex 2360  wss 2999   class class class wbr 3845  cima 4441  wf 5011  1-1-ontowf1o 5014  cfv 5015   Isom wiso 5016
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024
This theorem is referenced by: (None)
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