Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fnunsn | GIF version |
Description: Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
fnunop.x | ⊢ (𝜑 → 𝑋 ∈ V) |
fnunop.y | ⊢ (𝜑 → 𝑌 ∈ V) |
fnunop.f | ⊢ (𝜑 → 𝐹 Fn 𝐷) |
fnunop.g | ⊢ 𝐺 = (𝐹 ∪ {〈𝑋, 𝑌〉}) |
fnunop.e | ⊢ 𝐸 = (𝐷 ∪ {𝑋}) |
fnunop.d | ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷) |
Ref | Expression |
---|---|
fnunsn | ⊢ (𝜑 → 𝐺 Fn 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnunop.f | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐷) | |
2 | fnunop.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ V) | |
3 | fnunop.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ V) | |
4 | fnsng 4947 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {〈𝑋, 𝑌〉} Fn {𝑋}) | |
5 | 2, 3, 4 | syl2anc 391 | . . 3 ⊢ (𝜑 → {〈𝑋, 𝑌〉} Fn {𝑋}) |
6 | fnunop.d | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷) | |
7 | disjsn 3432 | . . . 4 ⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐷) | |
8 | 6, 7 | sylibr 137 | . . 3 ⊢ (𝜑 → (𝐷 ∩ {𝑋}) = ∅) |
9 | fnun 5005 | . . 3 ⊢ (((𝐹 Fn 𝐷 ∧ {〈𝑋, 𝑌〉} Fn {𝑋}) ∧ (𝐷 ∩ {𝑋}) = ∅) → (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) | |
10 | 1, 5, 8, 9 | syl21anc 1134 | . 2 ⊢ (𝜑 → (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) |
11 | fnunop.g | . . . 4 ⊢ 𝐺 = (𝐹 ∪ {〈𝑋, 𝑌〉}) | |
12 | 11 | fneq1i 4993 | . . 3 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn 𝐸) |
13 | fnunop.e | . . . 4 ⊢ 𝐸 = (𝐷 ∪ {𝑋}) | |
14 | 13 | fneq2i 4994 | . . 3 ⊢ ((𝐹 ∪ {〈𝑋, 𝑌〉}) Fn 𝐸 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) |
15 | 12, 14 | bitri 173 | . 2 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) |
16 | 10, 15 | sylibr 137 | 1 ⊢ (𝜑 → 𝐺 Fn 𝐸) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∪ cun 2915 ∩ cin 2916 ∅c0 3224 {csn 3375 〈cop 3378 Fn wfn 4897 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-fun 4904 df-fn 4905 |
This theorem is referenced by: tfrlemisucfn 5938 |
Copyright terms: Public domain | W3C validator |