Theorem elrnust 22918

Description: First direction for ustbas 22921. (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 6691	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ dom UnifOn)
2		fveq2 6659	. . . . 5 ⊢ (𝑥 = 𝑋 → (UnifOn‘𝑥) = (UnifOn‘𝑋))
3	2	eleq2d 2838	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋)))
4	3	rspcev 3542	. . 3 ⊢ ((𝑋 ∈ dom UnifOn ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
5	1, 4	mpancom 688	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
6		ustfn 22895	. . 3 ⊢ UnifOn Fn V
7		fnfun 6435	. . 3 ⊢ (UnifOn Fn V → Fun UnifOn)
8		elunirn 7003	. . 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 237	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1539   ∈ wcel 2112  ∃wrex 3072  Vcvv 3410  ∪ cuni 4799  dom cdm 5525  ran crn 5526  Fun wfun 6330   Fn wfn 6331  ‘cfv 6336  UnifOncust 22893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6295  df-fun 6338  df-fn 6339  df-fv 6344  df-ust 22894

This theorem is referenced by:  ustbas  22921  utopval  22926  tusval  22960  ucnval  22971  iscfilu  22982