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Theorem supisoex 6904
 Description: Lemma for supisoti 6905. (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 108 . . . . . 6 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
5 simpr 109 . . . . . 6 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → 𝐶𝐴)
64, 5supisolem 6903 . . . . 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 5716 . . . . . . . 8 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
8 f1of 5375 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
94, 7, 83syl 17 . . . . . . 7 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → 𝐹:𝐴𝐵)
109ffvelrnda 5563 . . . . . 6 (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
11 breq1 3940 . . . . . . . . . . 11 (𝑢 = (𝐹𝑥) → (𝑢𝑆𝑤 ↔ (𝐹𝑥)𝑆𝑤))
1211notbid 657 . . . . . . . . . 10 (𝑢 = (𝐹𝑥) → (¬ 𝑢𝑆𝑤 ↔ ¬ (𝐹𝑥)𝑆𝑤))
1312ralbidv 2438 . . . . . . . . 9 (𝑢 = (𝐹𝑥) → (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ↔ ∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤))
14 breq2 3941 . . . . . . . . . . 11 (𝑢 = (𝐹𝑥) → (𝑤𝑆𝑢𝑤𝑆(𝐹𝑥)))
1514imbi1d 230 . . . . . . . . . 10 (𝑢 = (𝐹𝑥) → ((𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣) ↔ (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
1615ralbidv 2438 . . . . . . . . 9 (𝑢 = (𝐹𝑥) → (∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣) ↔ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
1713, 16anbi12d 465 . . . . . . . 8 (𝑢 = (𝐹𝑥) → ((∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
1817rspcev 2793 . . . . . . 7 (((𝐹𝑥) ∈ 𝐵 ∧ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))) → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
1918ex 114 . . . . . 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 149 . . . 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 2552 . . 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 409 . 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 103   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418   ⊆ wss 3076   class class class wbr 3937   “ cima 4550  ⟶wf 5127  –1-1-onto→wf1o 5130  ‘cfv 5131   Isom wiso 5132 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140 This theorem is referenced by: (None)
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