Theorem elrnust 22825

Description: First direction for ustbas 22828. (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 6695	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ dom UnifOn)
2		fveq2 6663	. . . . 5 ⊢ (𝑥 = 𝑋 → (UnifOn‘𝑥) = (UnifOn‘𝑋))
3	2	eleq2d 2896	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋)))
4	3	rspcev 3621	. . 3 ⊢ ((𝑋 ∈ dom UnifOn ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
5	1, 4	mpancom 686	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
6		ustfn 22802	. . 3 ⊢ UnifOn Fn V
7		fnfun 6446	. . 3 ⊢ (UnifOn Fn V → Fun UnifOn)
8		elunirn 7002	. . 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 236	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1531   ∈ wcel 2108  ∃wrex 3137  Vcvv 3493  ∪ cuni 4830  dom cdm 5548  ran crn 5549  Fun wfun 6342   Fn wfn 6343  ‘cfv 6348  UnifOncust 22800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-ust 22801

This theorem is referenced by:  ustbas  22828  utopval  22833  tusval  22867  ucnval  22878  iscfilu  22889