Theorem supisoex 9504

Description: Lemma for supiso 9505. (Contributed by Mario Carneiro, 24-Dec-2016.)

Hypotheses

Ref	Expression
supiso.1	⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
supiso.2	⊢ (𝜑 → 𝐶 ⊆ 𝐴)
supisoex.3	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))

Assertion

Ref	Expression
supisoex	⊢ (𝜑 → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)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

Step	Hyp	Ref	Expression
1		supisoex.3	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))
2		supiso.1	. . 3 ⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
3		supiso.2	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
4		simpl 488	. . . . . 6 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
5		simpr 490	. . . . . 6 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
6	4, 5	supisolem 9503	. . . . 5 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝑥)𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝑥) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))
7		isof1o 7362	. . . . . . . 8 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴⟶1-1onto𝐵)
8		f1of 6854	. . . . . . . 8 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → 𝐹:𝐴⟶𝐵)
9	4, 7, 8	3syl 19	. . . . . . 7 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → 𝐹:𝐴⟶𝐵)
10	9	ffvelcdmda 7116	. . . . . 6 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
11		breq1 5111	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐹‘𝑥) → (𝑢𝑆𝑤 ↔ (𝐹‘𝑥)𝑆𝑤))
12	11	notbid 321	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘𝑥) → (¬ 𝑢𝑆𝑤 ↔ ¬ (𝐹‘𝑥)𝑆𝑤))
13	12	ralbidv 3189	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑥) → (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ↔ ∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝑥)𝑆𝑤))
14		breq2 5112	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐹‘𝑥) → (𝑤𝑆𝑢 ↔ 𝑤𝑆(𝐹‘𝑥)))
15	14	imbi1d 344	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘𝑥) → ((𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣) ↔ (𝑤𝑆(𝐹‘𝑥) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))
16	15	ralbidv 3189	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑥) → (∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣) ↔ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝑥) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))
17	13, 16	anbi12d 644	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑥) → ((∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝑥)𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝑥) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))
18	17	rspcev 3580	. . . . . . 7 ⊢ (((𝐹‘𝑥) ∈ 𝐵 ∧ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝑥)𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝑥) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))) → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))
19	18	ex 418	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ 𝐵 → ((∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝑥)𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝑥) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)) → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))
20	10, 19	syl 18	. . . . 5 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝑥)𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝑥) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)) → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))
21	6, 20	sylbid 243	. . . 4 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)) → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))
22	21	rexlimdva 3167	. . 3 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)) → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))
23	2, 3, 22	syl2anc 596	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)) → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))
24	1, 23	mpd 16	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   class class class wbr 5108   “ cima 5661  ⟶wf 6562  ⟶1-1ontowf1o 6565  ‘cfv 6566   Isom wiso 6567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575

This theorem is used by: (None)