![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mreunirn | Structured version Visualization version GIF version |
Description: Two ways to express the notion of being a Moore collection on an unspecified base. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
mreunirn | ⊢ (𝐶 ∈ ∪ ran Moore ↔ 𝐶 ∈ (Moore‘∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmre 16452 | . . . 4 ⊢ Moore Fn V | |
2 | fnunirn 6652 | . . . 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 16461 | . . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑥) → ∪ 𝐶 = 𝑥) | |
5 | 4 | fveq2d 6334 | . . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑥) → (Moore‘∪ 𝐶) = (Moore‘𝑥)) |
6 | 5 | eleq2d 2836 | . . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑥) → (𝐶 ∈ (Moore‘∪ 𝐶) ↔ 𝐶 ∈ (Moore‘𝑥))) |
7 | 6 | ibir 257 | . . . 4 ⊢ (𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘∪ 𝐶)) |
8 | 7 | rexlimivw 3177 | . . 3 ⊢ (∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘∪ 𝐶)) |
9 | 3, 8 | sylbi 207 | . 2 ⊢ (𝐶 ∈ ∪ ran Moore → 𝐶 ∈ (Moore‘∪ 𝐶)) |
10 | fvssunirn 6356 | . . 3 ⊢ (Moore‘∪ 𝐶) ⊆ ∪ ran Moore | |
11 | 10 | sseli 3748 | . 2 ⊢ (𝐶 ∈ (Moore‘∪ 𝐶) → 𝐶 ∈ ∪ ran Moore) |
12 | 9, 11 | impbii 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 |