Theorem ustuqtop3 24589

Description: Lemma for ustuqtop 24592, similar to elnei 23456. (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 6687	. . . . . . 7 ⊢ ( I ↾ 𝑋) Fn 𝑋
2		fnsnfv 6984	. . . . . . 7 ⊢ ((( I ↾ 𝑋) Fn 𝑋 ∧ 𝑝 ∈ 𝑋) → {(( I ↾ 𝑋)‘𝑝)} = (( I ↾ 𝑋) “ {𝑝}))
3	1, 2	mpan 703	. . . . . 6 ⊢ (𝑝 ∈ 𝑋 → {(( I ↾ 𝑋)‘𝑝)} = (( I ↾ 𝑋) “ {𝑝}))
4	3	ad4antlr 746	. . . . 5 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → {(( I ↾ 𝑋)‘𝑝)} = (( I ↾ 𝑋) “ {𝑝}))
5		ustdiag 24555	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑤)
6	5	ad5ant14 770	. . . . . 6 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ( I ↾ 𝑋) ⊆ 𝑤)
7		imass1 6065	. . . . . 6 ⊢ (( I ↾ 𝑋) ⊆ 𝑤 → (( I ↾ 𝑋) “ {𝑝}) ⊆ (𝑤 “ {𝑝}))
8	6, 7	syl 18	. . . . 5 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (( I ↾ 𝑋) “ {𝑝}) ⊆ (𝑤 “ {𝑝}))
9	4, 8	eqsstrd 3947	. . . 4 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → {(( I ↾ 𝑋)‘𝑝)} ⊆ (𝑤 “ {𝑝}))
10		fvex 6918	. . . . 5 ⊢ (( I ↾ 𝑋)‘𝑝) ∈ V
11	10	snss 4745	. . . 4 ⊢ ((( I ↾ 𝑋)‘𝑝) ∈ (𝑤 “ {𝑝}) ↔ {(( I ↾ 𝑋)‘𝑝)} ⊆ (𝑤 “ {𝑝}))
12	9, 11	sylibr 237	. . 3 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (( I ↾ 𝑋)‘𝑝) ∈ (𝑤 “ {𝑝}))
13		fvresi 7198	. . . . 5 ⊢ (𝑝 ∈ 𝑋 → (( I ↾ 𝑋)‘𝑝) = 𝑝)
14	13	eqcomd 2769	. . . 4 ⊢ (𝑝 ∈ 𝑋 → 𝑝 = (( I ↾ 𝑋)‘𝑝))
15	14	ad4antlr 746	. . 3 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑝 = (( I ↾ 𝑋)‘𝑝))
16		simpr 490	. . 3 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑎 = (𝑤 “ {𝑝}))
17	12, 15, 16	3eltr4d 2879	. 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 24585	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
20	19	elvd 3460	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
21	20	biimpa 482	. 2 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))
22	17, 21	r19.29a 3174	1 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  {csn 4588   ↦ cmpt 5189   I cid 5549  ran crn 5656   ↾ cres 5657   “ cima 5658   Fn wfn 6550  ‘cfv 6555  UnifOncust 24546

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-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

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-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ust 24547

This theorem is used by:  ustuqtop  24592  utopsnneiplem  24593