Theorem gruen 10804

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.)

Assertion

Ref	Expression
gruen	⊢ ((𝑈 ∈ Univ ∧ 𝐴 ⊆ 𝑈 ∧ (𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴)) → 𝐴 ∈ 𝑈)

Proof of Theorem gruen

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 8946	. . . . 5 ⊢ (𝐵 ≈ 𝐴 ↔ ∃𝑦 𝑦:𝐵–1-1-onto→𝐴)
2		f1ofo 6831	. . . . . . . . 9 ⊢ (𝑦:𝐵–1-1-onto→𝐴 → 𝑦:𝐵–onto→𝐴)
3		simp3l 1198	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ (𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈)) → 𝑦:𝐵–onto→𝐴)
4		forn 6799	. . . . . . . . . . . . 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 6796	. . . . . . . . . . . . . 14 ⊢ (𝑦:𝐵–onto→𝐴 → 𝑦:𝐵⟶𝐴)
7		fss 6725	. . . . . . . . . . . . . 14 ⊢ ((𝑦:𝐵⟶𝐴 ∧ 𝐴 ⊆ 𝑈) → 𝑦:𝐵⟶𝑈)
8	6, 7	sylan 579	. . . . . . . . . . . . 13 ⊢ ((𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈) → 𝑦:𝐵⟶𝑈)
9		grurn 10793	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ 𝑦:𝐵⟶𝑈) → ran 𝑦 ∈ 𝑈)
10	8, 9	syl3an3 1162	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ (𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈)) → ran 𝑦 ∈ 𝑈)
11	5, 10	eqeltrrd 2826	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ (𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈)) → 𝐴 ∈ 𝑈)
12	11	3expia 1118	. . . . . . . . . 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 1928	. . . . . . 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 1114	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ⊆ 𝑈 ∧ (𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴)) → 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098   ⊆ wss 3941   class class class wbr 5139  ran crn 5668  ⟶wf 6530  –onto→wfo 6532  –1-1-onto→wf1o 6533   ≈ cen 8933  Univcgru 10782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-en 8937  df-gru 10783

This theorem is referenced by:  grudomon  10809  gruina  10810