Theorem mreunirn 17522

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 17512	. . . 4 ⊢ Moore Fn V
2		fnunirn 7199	. . . 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 17521	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑥) → ∪ 𝐶 = 𝑥)
5	4	fveq2d 6837	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑥) → (Moore‘∪ 𝐶) = (Moore‘𝑥))
6	5	eleq2d 2821	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑥) → (𝐶 ∈ (Moore‘∪ 𝐶) ↔ 𝐶 ∈ (Moore‘𝑥)))
7	6	ibir 268	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘∪ 𝐶))
8	7	rexlimivw 3132	. . 3 ⊢ (∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘∪ 𝐶))
9	3, 8	sylbi 217	. 2 ⊢ (𝐶 ∈ ∪ ran Moore → 𝐶 ∈ (Moore‘∪ 𝐶))
10		fvssunirn 6864	. . 3 ⊢ (Moore‘∪ 𝐶) ⊆ ∪ ran Moore
11	10	sseli 3928	. 2 ⊢ (𝐶 ∈ (Moore‘∪ 𝐶) → 𝐶 ∈ ∪ ran Moore)
12	9, 11	impbii 209	1 ⊢ (𝐶 ∈ ∪ ran Moore ↔ 𝐶 ∈ (Moore‘∪ 𝐶))

Syntax hints:   ↔ wb 206   ∈ wcel 2114  ∃wrex 3059  Vcvv 3439  ∪ cuni 4862  ran crn 5624   Fn wfn 6486  ‘cfv 6491  Moorecmre 17503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6447  df-fun 6493  df-fn 6494  df-fv 6499  df-mre 17507

This theorem is referenced by:  fnmrc  17532  mrcfval  17533