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Theorem mreunirn 16189
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 16179 . . . 4 Moore Fn V
2 fnunirn 6471 . . . 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 16188 . . . . . . 7 (𝐶 ∈ (Moore‘𝑥) → 𝐶 = 𝑥)
54fveq2d 6157 . . . . . 6 (𝐶 ∈ (Moore‘𝑥) → (Moore‘ 𝐶) = (Moore‘𝑥))
65eleq2d 2684 . . . . 5 (𝐶 ∈ (Moore‘𝑥) → (𝐶 ∈ (Moore‘ 𝐶) ↔ 𝐶 ∈ (Moore‘𝑥)))
76ibir 257 . . . 4 (𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘ 𝐶))
87rexlimivw 3023 . . 3 (∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘ 𝐶))
93, 8sylbi 207 . 2 (𝐶 ran Moore → 𝐶 ∈ (Moore‘ 𝐶))
10 fvssunirn 6179 . . 3 (Moore‘ 𝐶) ⊆ ran Moore
1110sseli 3583 . 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 1987  wrex 2908  Vcvv 3189   cuni 4407  ran crn 5080   Fn wfn 5847  cfv 5852  Moorecmre 16170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860  df-mre 16174
This theorem is referenced by:  fnmrc  16195  mrcfval  16196
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