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Theorem supisoex 8340
Description: Lemma for supiso 8341. (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 473 . . . . . 6 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
5 simpr 477 . . . . . 6 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → 𝐶𝐴)
64, 5supisolem 8339 . . . . 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 6538 . . . . . . . 8 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
8 f1of 6104 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
94, 7, 83syl 18 . . . . . . 7 ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → 𝐹:𝐴𝐵)
109ffvelrnda 6325 . . . . . 6 (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
11 breq1 4626 . . . . . . . . . . 11 (𝑢 = (𝐹𝑥) → (𝑢𝑆𝑤 ↔ (𝐹𝑥)𝑆𝑤))
1211notbid 308 . . . . . . . . . 10 (𝑢 = (𝐹𝑥) → (¬ 𝑢𝑆𝑤 ↔ ¬ (𝐹𝑥)𝑆𝑤))
1312ralbidv 2982 . . . . . . . . 9 (𝑢 = (𝐹𝑥) → (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ↔ ∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤))
14 breq2 4627 . . . . . . . . . . 11 (𝑢 = (𝐹𝑥) → (𝑤𝑆𝑢𝑤𝑆(𝐹𝑥)))
1514imbi1d 331 . . . . . . . . . 10 (𝑢 = (𝐹𝑥) → ((𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣) ↔ (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
1615ralbidv 2982 . . . . . . . . 9 (𝑢 = (𝐹𝑥) → (∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣) ↔ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
1713, 16anbi12d 746 . . . . . . . 8 (𝑢 = (𝐹𝑥) → ((∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
1817rspcev 3299 . . . . . . 7 (((𝐹𝑥) ∈ 𝐵 ∧ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝑥)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝑥) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))) → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
1918ex 450 . . . . . 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 230 . . . 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 3026 . . 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 692 . 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 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909  wss 3560   class class class wbr 4623  cima 5087  wf 5853  1-1-ontowf1o 5856  cfv 5857   Isom wiso 5858
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 4751  ax-nul 4759  ax-pr 4877
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-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866
This theorem is referenced by: (None)
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