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Theorem wunf 10152
 Description: A weak universe is closed under functions with known domain and codomain. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
wunop.3 (𝜑𝐵𝑈)
wunf.3 (𝜑𝐹:𝐴𝐵)
Assertion
Ref Expression
wunf (𝜑𝐹𝑈)

Proof of Theorem wunf
StepHypRef Expression
1 wun0.1 . . 3 (𝜑𝑈 ∈ WUni)
2 wunop.3 . . . 4 (𝜑𝐵𝑈)
3 wunop.2 . . . 4 (𝜑𝐴𝑈)
41, 2, 3wunmap 10151 . . 3 (𝜑 → (𝐵m 𝐴) ∈ 𝑈)
51, 4wunelss 10133 . 2 (𝜑 → (𝐵m 𝐴) ⊆ 𝑈)
6 wunf.3 . . 3 (𝜑𝐹:𝐴𝐵)
72, 3elmapd 8423 . . 3 (𝜑 → (𝐹 ∈ (𝐵m 𝐴) ↔ 𝐹:𝐴𝐵))
86, 7mpbird 259 . 2 (𝜑𝐹 ∈ (𝐵m 𝐴))
95, 8sseldd 3971 1 (𝜑𝐹𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2113  ⟶wf 6354  (class class class)co 7159   ↑m cmap 8409  WUnicwun 10125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-map 8411  df-pm 8412  df-wun 10127 This theorem is referenced by:  wunndx  16507  wunnat  17229  catcoppccl  17371  catcfuccl  17372  catcxpccl  17460
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