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Theorem mreunirn 16872
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 16862 . . . 4 Moore Fn V
2 fnunirn 7004 . . . 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 16871 . . . . . . 7 (𝐶 ∈ (Moore‘𝑥) → 𝐶 = 𝑥)
54fveq2d 6665 . . . . . 6 (𝐶 ∈ (Moore‘𝑥) → (Moore‘ 𝐶) = (Moore‘𝑥))
65eleq2d 2901 . . . . 5 (𝐶 ∈ (Moore‘𝑥) → (𝐶 ∈ (Moore‘ 𝐶) ↔ 𝐶 ∈ (Moore‘𝑥)))
76ibir 271 . . . 4 (𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘ 𝐶))
87rexlimivw 3274 . . 3 (∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘ 𝐶))
93, 8sylbi 220 . 2 (𝐶 ran Moore → 𝐶 ∈ (Moore‘ 𝐶))
10 fvssunirn 6690 . . 3 (Moore‘ 𝐶) ⊆ ran Moore
1110sseli 3949 . 2 (𝐶 ∈ (Moore‘ 𝐶) → 𝐶 ran Moore)
129, 11impbii 212 1 (𝐶 ran Moore ↔ 𝐶 ∈ (Moore‘ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2115  wrex 3134  Vcvv 3480   cuni 4824  ran crn 5543   Fn wfn 6338  cfv 6343  Moorecmre 16853
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-mre 16857
This theorem is referenced by:  fnmrc  16878  mrcfval  16879
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