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Theorem mreunirn 17659
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.
StepHypRef Expression
1 fnmre 17649 . . . 4 Moore Fn V
2 fnunirn 7291 . . . 4 (Moore Fn V → (𝐶 ran Moore ↔ ∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥)))
31, 2ax-mp 5 . . 3 (𝐶 ran Moore ↔ ∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥))
4 mreuni 17658 . . . . . . 7 (𝐶 ∈ (Moore‘𝑥) → 𝐶 = 𝑥)
54fveq2d 6924 . . . . . 6 (𝐶 ∈ (Moore‘𝑥) → (Moore‘ 𝐶) = (Moore‘𝑥))
65eleq2d 2830 . . . . 5 (𝐶 ∈ (Moore‘𝑥) → (𝐶 ∈ (Moore‘ 𝐶) ↔ 𝐶 ∈ (Moore‘𝑥)))
76ibir 268 . . . 4 (𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘ 𝐶))
87rexlimivw 3157 . . 3 (∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘ 𝐶))
93, 8sylbi 217 . 2 (𝐶 ran Moore → 𝐶 ∈ (Moore‘ 𝐶))
10 fvssunirn 6953 . . 3 (Moore‘ 𝐶) ⊆ ran Moore
1110sseli 4004 . 2 (𝐶 ∈ (Moore‘ 𝐶) → 𝐶 ran Moore)
129, 11impbii 209 1 (𝐶 ran Moore ↔ 𝐶 ∈ (Moore‘ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2108  wrex 3076  Vcvv 3488   cuni 4931  ran crn 5701   Fn wfn 6568  cfv 6573  Moorecmre 17640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581  df-mre 17644
This theorem is referenced by:  fnmrc  17665  mrcfval  17666
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