Theorem ustuqtop3 24267

Description: Lemma for ustuqtop 24270, similar to elnei 23134. (Contributed by Thierry Arnoux, 11-Jan-2018.)

Hypothesis

Ref	Expression
utopustuq.1	⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))

Assertion

Ref	Expression
ustuqtop3	⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)

Distinct variable groups:   𝑣,𝑝,𝑈   𝑋,𝑝,𝑣,𝑎   𝑁,𝑎,𝑝   𝑣,𝑎,𝑈   𝑋,𝑎

Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnresi 6697	. . . . . . 7 ⊢ ( I ↾ 𝑋) Fn 𝑋
2		fnsnfv 6987	. . . . . . 7 ⊢ ((( I ↾ 𝑋) Fn 𝑋 ∧ 𝑝 ∈ 𝑋) → {(( I ↾ 𝑋)‘𝑝)} = (( I ↾ 𝑋) “ {𝑝}))
3	1, 2	mpan 690	. . . . . 6 ⊢ (𝑝 ∈ 𝑋 → {(( I ↾ 𝑋)‘𝑝)} = (( I ↾ 𝑋) “ {𝑝}))
4	3	ad4antlr 733	. . . . 5 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → {(( I ↾ 𝑋)‘𝑝)} = (( I ↾ 𝑋) “ {𝑝}))
5		ustdiag 24232	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑤)
6	5	ad5ant14 758	. . . . . 6 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ( I ↾ 𝑋) ⊆ 𝑤)
7		imass1 6121	. . . . . 6 ⊢ (( I ↾ 𝑋) ⊆ 𝑤 → (( I ↾ 𝑋) “ {𝑝}) ⊆ (𝑤 “ {𝑝}))
8	6, 7	syl 17	. . . . 5 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (( I ↾ 𝑋) “ {𝑝}) ⊆ (𝑤 “ {𝑝}))
9	4, 8	eqsstrd 4033	. . . 4 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → {(( I ↾ 𝑋)‘𝑝)} ⊆ (𝑤 “ {𝑝}))
10		fvex 6919	. . . . 5 ⊢ (( I ↾ 𝑋)‘𝑝) ∈ V
11	10	snss 4789	. . . 4 ⊢ ((( I ↾ 𝑋)‘𝑝) ∈ (𝑤 “ {𝑝}) ↔ {(( I ↾ 𝑋)‘𝑝)} ⊆ (𝑤 “ {𝑝}))
12	9, 11	sylibr 234	. . 3 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (( I ↾ 𝑋)‘𝑝) ∈ (𝑤 “ {𝑝}))
13		fvresi 7192	. . . . 5 ⊢ (𝑝 ∈ 𝑋 → (( I ↾ 𝑋)‘𝑝) = 𝑝)
14	13	eqcomd 2740	. . . 4 ⊢ (𝑝 ∈ 𝑋 → 𝑝 = (( I ↾ 𝑋)‘𝑝))
15	14	ad4antlr 733	. . 3 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑝 = (( I ↾ 𝑋)‘𝑝))
16		simpr 484	. . 3 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑎 = (𝑤 “ {𝑝}))
17	12, 15, 16	3eltr4d 2853	. 2 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑝 ∈ 𝑎)
18		utopustuq.1	. . . . 5 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
19	18	ustuqtoplem 24263	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
20	19	elvd 3483	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
21	20	biimpa 476	. 2 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))
22	17, 21	r19.29a 3159	1 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∃wrex 3067  Vcvv 3477   ⊆ wss 3962  {csn 4630   ↦ cmpt 5230   I cid 5581  ran crn 5689   ↾ cres 5690   “ cima 5691   Fn wfn 6557  ‘cfv 6562  UnifOncust 24223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ust 24224

This theorem is referenced by:  ustuqtop  24270  utopsnneiplem  24271