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Theorem gruf 9593
 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 1061 . . . 4 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹:𝐴𝑈)
21feqmptd 6216 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
3 fvex 6168 . . . 4 (𝐹𝑥) ∈ V
43fnasrn 6376 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) = ran (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩)
52, 4syl6eq 2671 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹 = ran (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩))
6 simpl1 1062 . . . . 5 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) ∧ 𝑥𝐴) → 𝑈 ∈ Univ)
7 gruel 9585 . . . . . . 7 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝑥𝐴) → 𝑥𝑈)
873expa 1262 . . . . . 6 (((𝑈 ∈ Univ ∧ 𝐴𝑈) ∧ 𝑥𝐴) → 𝑥𝑈)
983adantl3 1217 . . . . 5 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) ∧ 𝑥𝐴) → 𝑥𝑈)
10 ffvelrn 6323 . . . . . 6 ((𝐹:𝐴𝑈𝑥𝐴) → (𝐹𝑥) ∈ 𝑈)
11103ad2antl3 1223 . . . . 5 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝑈)
12 gruop 9587 . . . . 5 ((𝑈 ∈ Univ ∧ 𝑥𝑈 ∧ (𝐹𝑥) ∈ 𝑈) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝑈)
136, 9, 11, 12syl3anc 1323 . . . 4 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) ∧ 𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝑈)
14 eqid 2621 . . . 4 (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩) = (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩)
1513, 14fmptd 6351 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩):𝐴𝑈)
16 grurn 9583 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩):𝐴𝑈) → ran (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩) ∈ 𝑈)
1715, 16syld3an3 1368 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩) ∈ 𝑈)
185, 17eqeltrd 2698 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   ∈ wcel 1987  ⟨cop 4161   ↦ cmpt 4683  ran crn 5085  ⟶wf 5853  ‘cfv 5857  Univcgru 9572 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-gru 9573 This theorem is referenced by: (None)
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