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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 16714 | . . . 4 ⊢ Moore Fn V | |
2 | fnunirn 6835 | . . . 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 16723 | . . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑥) → ∪ 𝐶 = 𝑥) | |
5 | 4 | fveq2d 6501 | . . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑥) → (Moore‘∪ 𝐶) = (Moore‘𝑥)) |
6 | 5 | eleq2d 2848 | . . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑥) → (𝐶 ∈ (Moore‘∪ 𝐶) ↔ 𝐶 ∈ (Moore‘𝑥))) |
7 | 6 | ibir 260 | . . . 4 ⊢ (𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘∪ 𝐶)) |
8 | 7 | rexlimivw 3224 | . . 3 ⊢ (∃𝑥 ∈ V 𝐶 ∈ (Moore‘𝑥) → 𝐶 ∈ (Moore‘∪ 𝐶)) |
9 | 3, 8 | sylbi 209 | . 2 ⊢ (𝐶 ∈ ∪ ran Moore → 𝐶 ∈ (Moore‘∪ 𝐶)) |
10 | fvssunirn 6526 | . . 3 ⊢ (Moore‘∪ 𝐶) ⊆ ∪ ran Moore | |
11 | 10 | sseli 3853 | . 2 ⊢ (𝐶 ∈ (Moore‘∪ 𝐶) → 𝐶 ∈ ∪ ran Moore) |
12 | 9, 11 | impbii 201 | 1 ⊢ (𝐶 ∈ ∪ ran Moore ↔ 𝐶 ∈ (Moore‘∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2048 ∃wrex 3086 Vcvv 3412 ∪ cuni 4710 ran crn 5405 Fn wfn 6181 ‘cfv 6186 Moorecmre 16705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2747 ax-sep 5058 ax-nul 5065 ax-pow 5117 ax-pr 5184 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2756 df-cleq 2768 df-clel 2843 df-nfc 2915 df-ne 2965 df-ral 3090 df-rex 3091 df-rab 3094 df-v 3414 df-sbc 3681 df-dif 3831 df-un 3833 df-in 3835 df-ss 3842 df-nul 4178 df-if 4349 df-pw 4422 df-sn 4440 df-pr 4442 df-op 4446 df-uni 4711 df-br 4928 df-opab 4990 df-mpt 5007 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-iota 6150 df-fun 6188 df-fn 6189 df-fv 6194 df-mre 16709 |
This theorem is referenced by: fnmrc 16730 mrcfval 16731 |
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