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Theorem wunf 9751
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 9750 . . 3 (𝜑 → (𝐵𝑚 𝐴) ∈ 𝑈)
51, 4wunelss 9732 . 2 (𝜑 → (𝐵𝑚 𝐴) ⊆ 𝑈)
6 wunf.3 . . 3 (𝜑𝐹:𝐴𝐵)
72, 3elmapd 8023 . . 3 (𝜑 → (𝐹 ∈ (𝐵𝑚 𝐴) ↔ 𝐹:𝐴𝐵))
86, 7mpbird 247 . 2 (𝜑𝐹 ∈ (𝐵𝑚 𝐴))
95, 8sseldd 3753 1 (𝜑𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  wf 6027  (class class class)co 6793  𝑚 cmap 8009  WUnicwun 9724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011  df-pm 8012  df-wun 9726
This theorem is referenced by:  wunndx  16085  wunnat  16823  catcoppccl  16965  catcfuccl  16966  catcxpccl  17055
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