Theorem setsn0fun 13336

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	⊢ (𝜑 → 𝑆 Struct 𝑋)
setsn0fun.i	⊢ (𝜑 → 𝐼 ∈ 𝑈)
setsn0fun.e	⊢ (𝜑 → 𝐸 ∈ 𝑊)

Assertion

Ref	Expression
setsn0fun	⊢ (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))

Proof of Theorem setsn0fun

Step	Hyp	Ref	Expression
1		setsn0fun.s	. 2 ⊢ (𝜑 → 𝑆 Struct 𝑋)
2		structn0fun 13312	. . 3 ⊢ (𝑆 Struct 𝑋 → Fun (𝑆 ∖ {∅}))
3		setsn0fun.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑈)
4		setsn0fun.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
5		structex 13311	. . . . . . 7 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V)
6		setsfun0 13335	. . . . . . 7 ⊢ (((𝑆 ∈ V ∧ Fun (𝑆 ∖ {∅})) ∧ (𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊)) → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
7	5, 6	sylanl1 402	. . . . . 6 ⊢ (((𝑆 Struct 𝑋 ∧ Fun (𝑆 ∖ {∅})) ∧ (𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊)) → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
8	7	expcom 116	. . . . 5 ⊢ ((𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → ((𝑆 Struct 𝑋 ∧ Fun (𝑆 ∖ {∅})) → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅})))
9	3, 4, 8	syl2anc 411	. . . 4 ⊢ (𝜑 → ((𝑆 Struct 𝑋 ∧ Fun (𝑆 ∖ {∅})) → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅})))
10	9	com12 30	. . 3 ⊢ ((𝑆 Struct 𝑋 ∧ Fun (𝑆 ∖ {∅})) → (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅})))
11	2, 10	mpdan 421	. 2 ⊢ (𝑆 Struct 𝑋 → (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅})))
12	1, 11	mpcom 36	1 ⊢ (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2205  Vcvv 2815   ∖ cdif 3211  ∅c0 3512  {csn 3694  ⟨cop 3697   class class class wbr 4114  Fun wfun 5351  (class class class)co 6058   Struct cstr 13295   sSet csts 13297

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-res 4766  df-iota 5317  df-fun 5359  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-struct 13301  df-sets 13306

This theorem is referenced by:  setsvtx  16175  setsiedg  16176