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Theorem wunf 10227
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 10226 . . 3 (𝜑 → (𝐵m 𝐴) ∈ 𝑈)
51, 4wunelss 10208 . 2 (𝜑 → (𝐵m 𝐴) ⊆ 𝑈)
6 wunf.3 . . 3 (𝜑𝐹:𝐴𝐵)
72, 3elmapd 8451 . . 3 (𝜑 → (𝐹 ∈ (𝐵m 𝐴) ↔ 𝐹:𝐴𝐵))
86, 7mpbird 260 . 2 (𝜑𝐹 ∈ (𝐵m 𝐴))
95, 8sseldd 3878 1 (𝜑𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  wf 6335  (class class class)co 7170  m cmap 8437  WUnicwun 10200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-map 8439  df-pm 8440  df-wun 10202
This theorem is referenced by:  wunndx  16607  wunnat  17331  catcoppccl  17484  catcfuccl  17485  catcxpccl  17573
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