Theorem elrnust 22075

Description: First direction for ustbas 22078. (Contributed by Thierry Arnoux, 16-Nov-2017.)

Assertion

Ref	Expression
elrnust	⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)

Proof of Theorem elrnust

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6258	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ dom UnifOn)
2		fveq2 6229	. . . . 5 ⊢ (𝑥 = 𝑋 → (UnifOn‘𝑥) = (UnifOn‘𝑋))
3	2	eleq2d 2716	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋)))
4	3	rspcev 3340	. . 3 ⊢ ((𝑋 ∈ dom UnifOn ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
5	1, 4	mpancom 704	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
6		ustfn 22052	. . 3 ⊢ UnifOn Fn V
7		fnfun 6026	. . 3 ⊢ (UnifOn Fn V → Fun UnifOn)
8		elunirn 6549	. . 3 ⊢ (Fun UnifOn → (𝑈 ∈ ∪ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥)))
9	6, 7, 8	mp2b 10	. 2 ⊢ (𝑈 ∈ ∪ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
10	5, 9	sylibr 224	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1523   ∈ wcel 2030  ∃wrex 2942  Vcvv 3231  ∪ cuni 4468  dom cdm 5143  ran crn 5144  Fun wfun 5920   Fn wfn 5921  ‘cfv 5926  UnifOncust 22050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-ust 22051

This theorem is referenced by:  ustbas  22078  utopval  22083  tusval  22117  ucnval  22128  iscfilu  22139