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Theorem supisoex 8935
Description: Lemma for supiso 8936. (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 486 . . . . . 6 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
5 simpr 488 . . . . . 6 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → 𝐶𝐴)
64, 5supisolem 8934 . . . . 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 7069 . . . . . . . 8 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
8 f1of 6606 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
94, 7, 83syl 18 . . . . . . 7 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → 𝐹:𝐴𝐵)
109ffvelrnda 6842 . . . . . 6 (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
11 breq1 5055 . . . . . . . . . . 11 (𝑢 = (𝐹𝑥) → (𝑢𝑆𝑤 ↔ (𝐹𝑥)𝑆𝑤))
1211notbid 321 . . . . . . . . . 10 (𝑢 = (𝐹𝑥) → (¬ 𝑢𝑆𝑤 ↔ ¬ (𝐹𝑥)𝑆𝑤))
1312ralbidv 3192 . . . . . . . . 9 (𝑢 = (𝐹𝑥) → (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ↔ ∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤))
14 breq2 5056 . . . . . . . . . . 11 (𝑢 = (𝐹𝑥) → (𝑤𝑆𝑢𝑤𝑆(𝐹𝑥)))
1514imbi1d 345 . . . . . . . . . 10 (𝑢 = (𝐹𝑥) → ((𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣) ↔ (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
1615ralbidv 3192 . . . . . . . . 9 (𝑢 = (𝐹𝑥) → (∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣) ↔ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
1713, 16anbi12d 633 . . . . . . . 8 (𝑢 = (𝐹𝑥) → ((∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
1817rspcev 3609 . . . . . . 7 (((𝐹𝑥) ∈ 𝐵 ∧ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))) → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
1918ex 416 . . . . . 6 ((𝐹𝑥) ∈ 𝐵 → ((∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)) → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
2010, 19syl 17 . . . . 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 243 . . . 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 3276 . . 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 587 . 2 (𝜑 → (∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)) → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
241, 23mpd 15 1 (𝜑 → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  wrex 3134  wss 3919   class class class wbr 5052  cima 5545  wf 6339  1-1-ontowf1o 6342  cfv 6343   Isom wiso 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352
This theorem is referenced by: (None)
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