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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 12011 | . . 3 ⊢ (𝑆 Struct 𝑋 → Fun (𝑆 ∖ {∅})) | |
3 | setsn0fun.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑈) | |
4 | setsn0fun.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
5 | structex 12010 | . . . . . . 7 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
6 | setsfun0 12034 | . . . . . . 7 ⊢ (((𝑆 ∈ V ∧ Fun (𝑆 ∖ {∅})) ∧ (𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊)) → Fun ((𝑆 sSet 〈𝐼, 𝐸〉) ∖ {∅})) | |
7 | 5, 6 | sylanl1 400 | . . . . . 6 ⊢ (((𝑆 Struct 𝑋 ∧ Fun (𝑆 ∖ {∅})) ∧ (𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊)) → Fun ((𝑆 sSet 〈𝐼, 𝐸〉) ∖ {∅})) |
8 | 7 | expcom 115 | . . . . 5 ⊢ ((𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → ((𝑆 Struct 𝑋 ∧ Fun (𝑆 ∖ {∅})) → Fun ((𝑆 sSet 〈𝐼, 𝐸〉) ∖ {∅}))) |
9 | 3, 4, 8 | syl2anc 409 | . . . 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 418 | . 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 103 ∈ wcel 1481 Vcvv 2689 ∖ cdif 3073 ∅c0 3368 {csn 3532 〈cop 3535 class class class wbr 3937 Fun wfun 5125 (class class class)co 5782 Struct cstr 11994 sSet csts 11996 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-res 4559 df-iota 5096 df-fun 5133 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-struct 12000 df-sets 12005 |
This theorem is referenced by: (None) |
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