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Theorem setsn0fun 12517
Description: The value of the structure replacement function (without the empty set) is a function if the structure (without the empty set) is a function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
Hypotheses
Ref Expression
setsn0fun.s (𝜑𝑆 Struct 𝑋)
setsn0fun.i (𝜑𝐼𝑈)
setsn0fun.e (𝜑𝐸𝑊)
Assertion
Ref Expression
setsn0fun (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))

Proof of Theorem setsn0fun
StepHypRef Expression
1 setsn0fun.s . 2 (𝜑𝑆 Struct 𝑋)
2 structn0fun 12493 . . 3 (𝑆 Struct 𝑋 → Fun (𝑆 ∖ {∅}))
3 setsn0fun.i . . . . 5 (𝜑𝐼𝑈)
4 setsn0fun.e . . . . 5 (𝜑𝐸𝑊)
5 structex 12492 . . . . . . 7 (𝑆 Struct 𝑋𝑆 ∈ V)
6 setsfun0 12516 . . . . . . 7 (((𝑆 ∈ V ∧ Fun (𝑆 ∖ {∅})) ∧ (𝐼𝑈𝐸𝑊)) → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
75, 6sylanl1 402 . . . . . 6 (((𝑆 Struct 𝑋 ∧ Fun (𝑆 ∖ {∅})) ∧ (𝐼𝑈𝐸𝑊)) → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
87expcom 116 . . . . 5 ((𝐼𝑈𝐸𝑊) → ((𝑆 Struct 𝑋 ∧ Fun (𝑆 ∖ {∅})) → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅})))
93, 4, 8syl2anc 411 . . . 4 (𝜑 → ((𝑆 Struct 𝑋 ∧ Fun (𝑆 ∖ {∅})) → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅})))
109com12 30 . . 3 ((𝑆 Struct 𝑋 ∧ Fun (𝑆 ∖ {∅})) → (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅})))
112, 10mpdan 421 . 2 (𝑆 Struct 𝑋 → (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅})))
121, 11mpcom 36 1 (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2160  Vcvv 2752  cdif 3141  c0 3437  {csn 3607  cop 3610   class class class wbr 4018  Fun wfun 5225  (class class class)co 5891   Struct cstr 12476   sSet csts 12478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-res 4653  df-iota 5193  df-fun 5233  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-struct 12482  df-sets 12487
This theorem is referenced by: (None)
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