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Mirrors > Home > ILE Home > Th. List > setsn0fun | GIF version |
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.) |
Ref | Expression |
---|---|
setsn0fun.s | ⊢ (𝜑 → 𝑆 Struct 𝑋) |
setsn0fun.i | ⊢ (𝜑 → 𝐼 ∈ 𝑈) |
setsn0fun.e | ⊢ (𝜑 → 𝐸 ∈ 𝑊) |
Ref | Expression |
---|---|
setsn0fun | ⊢ (𝜑 → Fun ((𝑆 sSet 〈𝐼, 𝐸〉) ∖ {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsn0fun.s | . 2 ⊢ (𝜑 → 𝑆 Struct 𝑋) | |
2 | structn0fun 12493 | . . 3 ⊢ (𝑆 Struct 𝑋 → Fun (𝑆 ∖ {∅})) | |
3 | setsn0fun.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑈) | |
4 | setsn0fun.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
5 | structex 12492 | . . . . . . 7 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
6 | setsfun0 12516 | . . . . . . 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〉) ∖ {∅})) |
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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