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Theorem gruf 10033
Description: A Grothendieck universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
gruf ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹𝑈)

Proof of Theorem gruf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp3 1118 . . . 4 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹:𝐴𝑈)
21feqmptd 6564 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
3 fvex 6514 . . . 4 (𝐹𝑥) ∈ V
43fnasrn 6732 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) = ran (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩)
52, 4syl6eq 2830 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹 = ran (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩))
6 simpl1 1171 . . . . 5 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) ∧ 𝑥𝐴) → 𝑈 ∈ Univ)
7 gruel 10025 . . . . . . 7 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝑥𝐴) → 𝑥𝑈)
873expa 1098 . . . . . 6 (((𝑈 ∈ Univ ∧ 𝐴𝑈) ∧ 𝑥𝐴) → 𝑥𝑈)
983adantl3 1148 . . . . 5 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) ∧ 𝑥𝐴) → 𝑥𝑈)
10 ffvelrn 6676 . . . . . 6 ((𝐹:𝐴𝑈𝑥𝐴) → (𝐹𝑥) ∈ 𝑈)
11103ad2antl3 1167 . . . . 5 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝑈)
12 gruop 10027 . . . . 5 ((𝑈 ∈ Univ ∧ 𝑥𝑈 ∧ (𝐹𝑥) ∈ 𝑈) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝑈)
136, 9, 11, 12syl3anc 1351 . . . 4 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) ∧ 𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝑈)
1413fmpttd 6704 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩):𝐴𝑈)
15 grurn 10023 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩):𝐴𝑈) → ran (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩) ∈ 𝑈)
1614, 15syld3an3 1389 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩) ∈ 𝑈)
175, 16eqeltrd 2866 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068  wcel 2050  cop 4448  cmpt 5009  ran crn 5409  wf 6186  cfv 6190  Univcgru 10012
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 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281
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 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-map 8210  df-gru 10013
This theorem is referenced by: (None)
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