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| 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 17634 | . . . 4 ⊢ Moore Fn V | |
| 2 | fnunirn 7274 | . . . 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 17643 | . . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑥) → ∪ 𝐶 = 𝑥) | |
| 5 | 4 | fveq2d 6910 | . . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑥) → (Moore‘∪ 𝐶) = (Moore‘𝑥)) |
| 6 | 5 | eleq2d 2827 | . . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑥) → (𝐶 ∈ (Moore‘∪ 𝐶) ↔ 𝐶 ∈ (Moore‘𝑥))) |
| 7 | 6 | ibir 268 | . . . 4 ⊢ (𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘∪ 𝐶)) |
| 8 | 7 | rexlimivw 3151 | . . 3 ⊢ (∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘∪ 𝐶)) |
| 9 | 3, 8 | sylbi 217 | . 2 ⊢ (𝐶 ∈ ∪ ran Moore → 𝐶 ∈ (Moore‘∪ 𝐶)) |
| 10 | fvssunirn 6939 | . . 3 ⊢ (Moore‘∪ 𝐶) ⊆ ∪ ran Moore | |
| 11 | 10 | sseli 3979 | . 2 ⊢ (𝐶 ∈ (Moore‘∪ 𝐶) → 𝐶 ∈ ∪ ran Moore) |
| 12 | 9, 11 | impbii 209 | 1 ⊢ (𝐶 ∈ ∪ ran Moore ↔ 𝐶 ∈ (Moore‘∪ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∪ cuni 4907 ran crn 5686 Fn wfn 6556 ‘cfv 6561 Moorecmre 17625 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-mre 17629 |
| This theorem is referenced by: fnmrc 17650 mrcfval 17651 |
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