Theorem mreunirn 16724

Description: Two ways to express the notion of being a Moore collection on an unspecified base. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Assertion

Ref	Expression
mreunirn	⊢ (𝐶 ∈ ∪ ran Moore ↔ 𝐶 ∈ (Moore‘∪ 𝐶))

Proof of Theorem mreunirn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnmre 16714	. . . 4 ⊢ Moore Fn V
2		fnunirn 6835	. . . 4 ⊢ (Moore Fn V → (𝐶 ∈ ∪ ran Moore ↔ ∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥)))
3	1, 2	ax-mp 5	. . 3 ⊢ (𝐶 ∈ ∪ ran Moore ↔ ∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥))
4		mreuni 16723	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑥) → ∪ 𝐶 = 𝑥)
5	4	fveq2d 6501	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑥) → (Moore‘∪ 𝐶) = (Moore‘𝑥))
6	5	eleq2d 2848	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑥) → (𝐶 ∈ (Moore‘∪ 𝐶) ↔ 𝐶 ∈ (Moore‘𝑥)))
7	6	ibir 260	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘∪ 𝐶))
8	7	rexlimivw 3224	. . 3 ⊢ (∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘∪ 𝐶))
9	3, 8	sylbi 209	. 2 ⊢ (𝐶 ∈ ∪ ran Moore → 𝐶 ∈ (Moore‘∪ 𝐶))
10		fvssunirn 6526	. . 3 ⊢ (Moore‘∪ 𝐶) ⊆ ∪ ran Moore
11	10	sseli 3853	. 2 ⊢ (𝐶 ∈ (Moore‘∪ 𝐶) → 𝐶 ∈ ∪ ran Moore)
12	9, 11	impbii 201	1 ⊢ (𝐶 ∈ ∪ ran Moore ↔ 𝐶 ∈ (Moore‘∪ 𝐶))

Syntax hints:   ↔ wb 198   ∈ wcel 2048  ∃wrex 3086  Vcvv 3412  ∪ cuni 4710  ran crn 5405   Fn wfn 6181  ‘cfv 6186  Moorecmre 16705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-sep 5058  ax-nul 5065  ax-pow 5117  ax-pr 5184

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ne 2965  df-ral 3090  df-rex 3091  df-rab 3094  df-v 3414  df-sbc 3681  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-nul 4178  df-if 4349  df-pw 4422  df-sn 4440  df-pr 4442  df-op 4446  df-uni 4711  df-br 4928  df-opab 4990  df-mpt 5007  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-iota 6150  df-fun 6188  df-fn 6189  df-fv 6194  df-mre 16709

This theorem is referenced by:  fnmrc  16730  mrcfval  16731