MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mreunirn Structured version   Visualization version   GIF version

Theorem mreunirn 16462
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 16452 . . . 4 Moore Fn V
2 fnunirn 6652 . . . 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 16461 . . . . . . 7 (𝐶 ∈ (Moore‘𝑥) → 𝐶 = 𝑥)
54fveq2d 6334 . . . . . 6 (𝐶 ∈ (Moore‘𝑥) → (Moore‘ 𝐶) = (Moore‘𝑥))
65eleq2d 2836 . . . . 5 (𝐶 ∈ (Moore‘𝑥) → (𝐶 ∈ (Moore‘ 𝐶) ↔ 𝐶 ∈ (Moore‘𝑥)))
76ibir 257 . . . 4 (𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘ 𝐶))
87rexlimivw 3177 . . 3 (∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘ 𝐶))
93, 8sylbi 207 . 2 (𝐶 ran Moore → 𝐶 ∈ (Moore‘ 𝐶))
10 fvssunirn 6356 . . 3 (Moore‘ 𝐶) ⊆ ran Moore
1110sseli 3748 . 2 (𝐶 ∈ (Moore‘ 𝐶) → 𝐶 ran Moore)
129, 11impbii 199 1 (𝐶 ran Moore ↔ 𝐶 ∈ (Moore‘ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2145  wrex 3062  Vcvv 3351   cuni 4574  ran crn 5250   Fn wfn 6024  cfv 6029  Moorecmre 16443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5992  df-fun 6031  df-fn 6032  df-fv 6037  df-mre 16447
This theorem is referenced by:  fnmrc  16468  mrcfval  16469
  Copyright terms: Public domain W3C validator