Theorem gruen 10223

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 8501	. . . . 5 ⊢ (𝐵 ≈ 𝐴 ↔ ∃𝑦 𝑦:𝐵–1-1-onto→𝐴)
2		f1ofo 6597	. . . . . . . . 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 6568	. . . . . . . . . . . . 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 6565	. . . . . . . . . . . . . 14 ⊢ (𝑦:𝐵–onto→𝐴 → 𝑦:𝐵⟶𝐴)
7		fss 6501	. . . . . . . . . . . . . 14 ⊢ ((𝑦:𝐵⟶𝐴 ∧ 𝐴 ⊆ 𝑈) → 𝑦:𝐵⟶𝑈)
8	6, 7	sylan 583	. . . . . . . . . . . . 13 ⊢ ((𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈) → 𝑦:𝐵⟶𝑈)
9		grurn 10212	. . . . . . . . . . . . 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 2891	. . . . . . . . . . 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 419	. . . . . . . . 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 1934	. . . . . . 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 456	. . . . 5 ⊢ ((𝐴 ⊆ 𝑈 ∧ 𝑈 ∈ Univ) → (𝐵 ∈ 𝑈 → (∃𝑦 𝑦:𝐵–1-1-onto→𝐴 → 𝐴 ∈ 𝑈)))
18	1, 17	syl7bi 258	. . . 4 ⊢ ((𝐴 ⊆ 𝑈 ∧ 𝑈 ∈ Univ) → (𝐵 ∈ 𝑈 → (𝐵 ≈ 𝐴 → 𝐴 ∈ 𝑈)))
19	18	impd 414	. . 3 ⊢ ((𝐴 ⊆ 𝑈 ∧ 𝑈 ∈ Univ) → ((𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴) → 𝐴 ∈ 𝑈))
20	19	ancoms 462	. 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 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ⊆ wss 3881   class class class wbr 5030  ran crn 5520  ⟶wf 6320  –onto→wfo 6322  –1-1-onto→wf1o 6323   ≈ cen 8489  Univcgru 10201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-en 8493  df-gru 10202

This theorem is referenced by:  grudomon  10228  gruina  10229