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| Mirrors > Home > MPE Home > Th. List > gruen | Structured version Visualization version GIF version | ||
| Description: A Grothendieck universe contains all subsets of itself that are equipotent to an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| Ref | Expression |
|---|---|
| gruen | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ⊆ 𝑈 ∧ (𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴)) → 𝐴 ∈ 𝑈) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bren 8995 | . . . . 5 ⊢ (𝐵 ≈ 𝐴 ↔ ∃𝑦 𝑦:𝐵–1-1-onto→𝐴) | |
| 2 | f1ofo 6855 | . . . . . . . . 9 ⊢ (𝑦:𝐵–1-1-onto→𝐴 → 𝑦:𝐵–onto→𝐴) | |
| 3 | simp3l 1202 | . . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ (𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈)) → 𝑦:𝐵–onto→𝐴) | |
| 4 | forn 6823 | . . . . . . . . . . . . 13 ⊢ (𝑦:𝐵–onto→𝐴 → ran 𝑦 = 𝐴) | |
| 5 | 3, 4 | syl 17 | . . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ (𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈)) → ran 𝑦 = 𝐴) |
| 6 | fof 6820 | . . . . . . . . . . . . . 14 ⊢ (𝑦:𝐵–onto→𝐴 → 𝑦:𝐵⟶𝐴) | |
| 7 | fss 6752 | . . . . . . . . . . . . . 14 ⊢ ((𝑦:𝐵⟶𝐴 ∧ 𝐴 ⊆ 𝑈) → 𝑦:𝐵⟶𝑈) | |
| 8 | 6, 7 | sylan 580 | . . . . . . . . . . . . 13 ⊢ ((𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈) → 𝑦:𝐵⟶𝑈) |
| 9 | grurn 10841 | . . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ 𝑦:𝐵⟶𝑈) → ran 𝑦 ∈ 𝑈) | |
| 10 | 8, 9 | syl3an3 1166 | . . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ (𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈)) → ran 𝑦 ∈ 𝑈) |
| 11 | 5, 10 | eqeltrrd 2842 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ (𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈)) → 𝐴 ∈ 𝑈) |
| 12 | 11 | 3expia 1122 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → ((𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈) → 𝐴 ∈ 𝑈)) |
| 13 | 12 | expd 415 | . . . . . . . . 9 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑦:𝐵–onto→𝐴 → (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈))) |
| 14 | 2, 13 | syl5 34 | . . . . . . . 8 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (𝑦:𝐵–1-1-onto→𝐴 → (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈))) |
| 15 | 14 | exlimdv 1933 | . . . . . . 7 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (∃𝑦 𝑦:𝐵–1-1-onto→𝐴 → (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈))) |
| 16 | 15 | com3r 87 | . . . . . 6 ⊢ (𝐴 ⊆ 𝑈 → ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈) → (∃𝑦 𝑦:𝐵–1-1-onto→𝐴 → 𝐴 ∈ 𝑈))) |
| 17 | 16 | expdimp 452 | . . . . 5 ⊢ ((𝐴 ⊆ 𝑈 ∧ 𝑈 ∈ Univ) → (𝐵 ∈ 𝑈 → (∃𝑦 𝑦:𝐵–1-1-onto→𝐴 → 𝐴 ∈ 𝑈))) |
| 18 | 1, 17 | syl7bi 255 | . . . 4 ⊢ ((𝐴 ⊆ 𝑈 ∧ 𝑈 ∈ Univ) → (𝐵 ∈ 𝑈 → (𝐵 ≈ 𝐴 → 𝐴 ∈ 𝑈))) |
| 19 | 18 | impd 410 | . . 3 ⊢ ((𝐴 ⊆ 𝑈 ∧ 𝑈 ∈ Univ) → ((𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴) → 𝐴 ∈ 𝑈)) |
| 20 | 19 | ancoms 458 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ⊆ 𝑈) → ((𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴) → 𝐴 ∈ 𝑈)) |
| 21 | 20 | 3impia 1118 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ⊆ 𝑈 ∧ (𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴)) → 𝐴 ∈ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 ran crn 5686 ⟶wf 6557 –onto→wfo 6559 –1-1-onto→wf1o 6560 ≈ cen 8982 Univcgru 10830 |
| 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 ax-un 7755 |
| 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 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-tr 5260 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-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-en 8986 df-gru 10831 |
| This theorem is referenced by: grudomon 10857 gruina 10858 |
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