Theorem ustuqtop5 24305

Description: Lemma for ustuqtop 24306. (Contributed by Thierry Arnoux, 11-Jan-2018.)

Hypothesis

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

Assertion

Ref	Expression
ustuqtop5	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))

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

Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop5

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ustbasel 24267	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
2		snssi 4744	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ⊆ 𝑋)
3		dfss 3923	. . . . . . . . 9 ⊢ ({𝑝} ⊆ 𝑋 ↔ {𝑝} = ({𝑝} ∩ 𝑋))
4	2, 3	sylib 220	. . . . . . . 8 ⊢ (𝑝 ∈ 𝑋 → {𝑝} = ({𝑝} ∩ 𝑋))
5		incom 4161	. . . . . . . 8 ⊢ ({𝑝} ∩ 𝑋) = (𝑋 ∩ {𝑝})
6	4, 5	eqtr2di 2814	. . . . . . 7 ⊢ (𝑝 ∈ 𝑋 → (𝑋 ∩ {𝑝}) = {𝑝})
7		snnzg 4733	. . . . . . 7 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ≠ ∅)
8	6, 7	eqnetrd 3024	. . . . . 6 ⊢ (𝑝 ∈ 𝑋 → (𝑋 ∩ {𝑝}) ≠ ∅)
9	8	adantl 485	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑋 ∩ {𝑝}) ≠ ∅)
10		xpima2 6170	. . . . 5 ⊢ ((𝑋 ∩ {𝑝}) ≠ ∅ → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋)
11	9, 10	syl 17	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋)
12	11	eqcomd 2768	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 = ((𝑋 × 𝑋) “ {𝑝}))
13		imaeq1 6044	. . . 4 ⊢ (𝑤 = (𝑋 × 𝑋) → (𝑤 “ {𝑝}) = ((𝑋 × 𝑋) “ {𝑝}))
14	13	rspceeqv 3604	. . 3 ⊢ (((𝑋 × 𝑋) ∈ 𝑈 ∧ 𝑋 = ((𝑋 × 𝑋) “ {𝑝})) → ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝}))
15	1, 12, 14	syl2an2r 695	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝}))
16		elfvex 6902	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
17		utopustuq.1	. . . 4 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
18	17	ustuqtoplem 24299	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑋 ∈ V) → (𝑋 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝})))
19	16, 18	mpidan 699	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑋 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝})))
20	15, 19	mpbird 259	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3086  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4582   ↦ cmpt 5181   × cxp 5645  ran crn 5648   “ cima 5650  ‘cfv 6521  UnifOncust 24260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ust 24261

This theorem is referenced by:  ustuqtop  24306  utopsnneiplem  24307