Theorem gruen 10849

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 8993	. . . . 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 1200	. . . . . . . . . . . . 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 10838	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ 𝑦:𝐵⟶𝑈) → ran 𝑦 ∈ 𝑈)
10	8, 9	syl3an3 1164	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ (𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈)) → ran 𝑦 ∈ 𝑈)
11	5, 10	eqeltrrd 2839	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ (𝑦:𝐵–onto→𝐴 ∧ 𝐴 ⊆ 𝑈)) → 𝐴 ∈ 𝑈)
12	11	3expia 1120	. . . . . . . . . 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 1930	. . . . . . 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 1116	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ⊆ 𝑈 ∧ (𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴)) → 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1536  ∃wex 1775   ∈ wcel 2105   ⊆ wss 3962   class class class wbr 5147  ran crn 5689  ⟶wf 6558  –onto→wfo 6560  –1-1-onto→wf1o 6561   ≈ cen 8980  Univcgru 10827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-en 8984  df-gru 10828

This theorem is referenced by:  grudomon  10854  gruina  10855