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Theorem gruf 10226
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 1135 . . . 4 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹:𝐴𝑈)
21feqmptd 6712 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
3 fvex 6662 . . . 4 (𝐹𝑥) ∈ V
43fnasrn 6888 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) = ran (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩)
52, 4eqtrdi 2852 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹 = ran (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩))
6 simpl1 1188 . . . . 5 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) ∧ 𝑥𝐴) → 𝑈 ∈ Univ)
7 gruel 10218 . . . . . . 7 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝑥𝐴) → 𝑥𝑈)
873expa 1115 . . . . . 6 (((𝑈 ∈ Univ ∧ 𝐴𝑈) ∧ 𝑥𝐴) → 𝑥𝑈)
983adantl3 1165 . . . . 5 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) ∧ 𝑥𝐴) → 𝑥𝑈)
10 ffvelrn 6830 . . . . . 6 ((𝐹:𝐴𝑈𝑥𝐴) → (𝐹𝑥) ∈ 𝑈)
11103ad2antl3 1184 . . . . 5 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝑈)
12 gruop 10220 . . . . 5 ((𝑈 ∈ Univ ∧ 𝑥𝑈 ∧ (𝐹𝑥) ∈ 𝑈) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝑈)
136, 9, 11, 12syl3anc 1368 . . . 4 (((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) ∧ 𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝑈)
1413fmpttd 6860 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩):𝐴𝑈)
15 grurn 10216 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩):𝐴𝑈) → ran (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩) ∈ 𝑈)
1614, 15syld3an3 1406 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran (𝑥𝐴 ↦ ⟨𝑥, (𝐹𝑥)⟩) ∈ 𝑈)
175, 16eqeltrd 2893 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2112  cop 4534  cmpt 5113  ran crn 5524  wf 6324  cfv 6328  Univcgru 10205
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-gru 10206
This theorem is referenced by: (None)
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