Theorem supisoex 9465

Description: Lemma for supiso 9466. (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 483	. . . . . 6 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
5		simpr 485	. . . . . 6 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
6	4, 5	supisolem 9464	. . . . 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 7316	. . . . . . . 8 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
8		f1of 6830	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
9	4, 7, 8	3syl 18	. . . . . . 7 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) → 𝐹:𝐴⟶𝐵)
10	9	ffvelcdmda 7083	. . . . . 6 ⊢ (((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
11		breq1 5150	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐹‘𝑥) → (𝑢𝑆𝑤 ↔ (𝐹‘𝑥)𝑆𝑤))
12	11	notbid 317	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘𝑥) → (¬ 𝑢𝑆𝑤 ↔ ¬ (𝐹‘𝑥)𝑆𝑤))
13	12	ralbidv 3177	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑥) → (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ↔ ∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝑥)𝑆𝑤))
14		breq2 5151	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐹‘𝑥) → (𝑤𝑆𝑢 ↔ 𝑤𝑆(𝐹‘𝑥)))
15	14	imbi1d 341	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘𝑥) → ((𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣) ↔ (𝑤𝑆(𝐹‘𝑥) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))
16	15	ralbidv 3177	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑥) → (∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣) ↔ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝑥) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))
17	13, 16	anbi12d 631	. . . . . . . 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 3612	. . . . . . 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 413	. . . . . 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 17	. . . . 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 239	. . . 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 3155	. . 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 584	. 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 15	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070   ⊆ wss 3947   class class class wbr 5147   “ cima 5678  ⟶wf 6536  –1-1-onto→wf1o 6539  ‘cfv 6540   Isom wiso 6541

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549

This theorem is referenced by: (None)