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Theorem setsn0fun 12010
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 11986 . . 3 (𝑆 Struct 𝑋 → Fun (𝑆 ∖ {∅}))
3 setsn0fun.i . . . . 5 (𝜑𝐼𝑈)
4 setsn0fun.e . . . . 5 (𝜑𝐸𝑊)
5 structex 11985 . . . . . . 7 (𝑆 Struct 𝑋𝑆 ∈ V)
6 setsfun0 12009 . . . . . . 7 (((𝑆 ∈ V ∧ Fun (𝑆 ∖ {∅})) ∧ (𝐼𝑈𝐸𝑊)) → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
75, 6sylanl1 399 . . . . . 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 408 . . . 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 417 . 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 1480  Vcvv 2686  cdif 3068  c0 3363  {csn 3527  cop 3530   class class class wbr 3929  Fun wfun 5117  (class class class)co 5774   Struct cstr 11969   sSet csts 11971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-struct 11975  df-sets 11980
This theorem is referenced by: (None)
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