![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 11672 | . . 3 ⊢ (𝑆 Struct 𝑋 → Fun (𝑆 ∖ {∅})) | |
3 | setsn0fun.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑈) | |
4 | setsn0fun.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
5 | structex 11671 | . . . . . . 7 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
6 | setsfun0 11695 | . . . . . . 7 ⊢ (((𝑆 ∈ V ∧ Fun (𝑆 ∖ {∅})) ∧ (𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊)) → Fun ((𝑆 sSet 〈𝐼, 𝐸〉) ∖ {∅})) | |
7 | 5, 6 | sylanl1 395 | . . . . . 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 404 | . . . 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 413 | . 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 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) |
Copyright terms: Public domain | W3C validator |