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Theorem wunf 10671
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 10670 . . 3 (𝜑 → (𝐵m 𝐴) ∈ 𝑈)
51, 4wunelss 10652 . 2 (𝜑 → (𝐵m 𝐴) ⊆ 𝑈)
6 wunf.3 . . 3 (𝜑𝐹:𝐴𝐵)
72, 3elmapd 8785 . . 3 (𝜑 → (𝐹 ∈ (𝐵m 𝐴) ↔ 𝐹:𝐴𝐵))
86, 7mpbird 257 . 2 (𝜑𝐹 ∈ (𝐵m 𝐴))
95, 8sseldd 3949 1 (𝜑𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wf 6496  (class class class)co 7361  m cmap 8771  WUnicwun 10644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-map 8773  df-pm 8774  df-wun 10646
This theorem is referenced by:  wunndx  17075  wunnat  17851  wunnatOLD  17852  catcoppccl  18011  catcoppcclOLD  18012  catcfuccl  18013  catcfucclOLD  18014  catcxpccl  18103  catcxpcclOLD  18104
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