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Theorem setsn0fun 11696
 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 11672 . . 3 (𝑆 Struct 𝑋 → Fun (𝑆 ∖ {∅}))
3 setsn0fun.i . . . . 5 (𝜑𝐼𝑈)
4 setsn0fun.e . . . . 5 (𝜑𝐸𝑊)
5 structex 11671 . . . . . . 7 (𝑆 Struct 𝑋𝑆 ∈ V)
6 setsfun0 11695 . . . . . . 7 (((𝑆 ∈ V ∧ Fun (𝑆 ∖ {∅})) ∧ (𝐼𝑈𝐸𝑊)) → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
75, 6sylanl1 395 . . . . . 6 (((𝑆 Struct 𝑋 ∧ Fun (𝑆 ∖ {∅})) ∧ (𝐼𝑈𝐸𝑊)) → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
87expcom 115 . . . . 5 ((𝐼𝑈𝐸𝑊) → ((𝑆 Struct 𝑋 ∧ Fun (𝑆 ∖ {∅})) → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅})))
93, 4, 8syl2anc 404 . . . 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 413 . 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 103   ∈ wcel 1445  Vcvv 2633   ∖ cdif 3010  ∅c0 3302  {csn 3466  ⟨cop 3469   class class class wbr 3867  Fun wfun 5043  (class class class)co 5690   Struct cstr 11655   sSet csts 11657 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-res 4479  df-iota 5014  df-fun 5051  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-struct 11661  df-sets 11666 This theorem is referenced by: (None)
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