Theorem mreunirn 17503

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 17493	. . . 4 ⊢ Moore Fn V
2		fnunirn 7187	. . . 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 17502	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑥) → ∪ 𝐶 = 𝑥)
5	4	fveq2d 6826	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑥) → (Moore‘∪ 𝐶) = (Moore‘𝑥))
6	5	eleq2d 2817	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑥) → (𝐶 ∈ (Moore‘∪ 𝐶) ↔ 𝐶 ∈ (Moore‘𝑥)))
7	6	ibir 268	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘∪ 𝐶))
8	7	rexlimivw 3129	. . 3 ⊢ (∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘∪ 𝐶))
9	3, 8	sylbi 217	. 2 ⊢ (𝐶 ∈ ∪ ran Moore → 𝐶 ∈ (Moore‘∪ 𝐶))
10		fvssunirn 6853	. . 3 ⊢ (Moore‘∪ 𝐶) ⊆ ∪ ran Moore
11	10	sseli 3925	. 2 ⊢ (𝐶 ∈ (Moore‘∪ 𝐶) → 𝐶 ∈ ∪ ran Moore)
12	9, 11	impbii 209	1 ⊢ (𝐶 ∈ ∪ ran Moore ↔ 𝐶 ∈ (Moore‘∪ 𝐶))

Syntax hints:   ↔ wb 206   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436  ∪ cuni 4856  ran crn 5615   Fn wfn 6476  ‘cfv 6481  Moorecmre 17484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489  df-mre 17488

This theorem is referenced by:  fnmrc  17513  mrcfval  17514