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 5170 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {〈𝑋, 𝑌〉} Fn {𝑋}) | |
5 | 2, 3, 4 | syl2anc 408 | . . 3 ⊢ (𝜑 → {〈𝑋, 𝑌〉} Fn {𝑋}) |
6 | fnunop.d | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷) | |
7 | disjsn 3585 | . . . 4 ⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐷) | |
8 | 6, 7 | sylibr 133 | . . 3 ⊢ (𝜑 → (𝐷 ∩ {𝑋}) = ∅) |
9 | fnun 5229 | . . 3 ⊢ (((𝐹 Fn 𝐷 ∧ {〈𝑋, 𝑌〉} Fn {𝑋}) ∧ (𝐷 ∩ {𝑋}) = ∅) → (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) | |
10 | 1, 5, 8, 9 | syl21anc 1215 | . 2 ⊢ (𝜑 → (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) |
11 | fnunop.g | . . . 4 ⊢ 𝐺 = (𝐹 ∪ {〈𝑋, 𝑌〉}) | |
12 | 11 | fneq1i 5217 | . . 3 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn 𝐸) |
13 | fnunop.e | . . . 4 ⊢ 𝐸 = (𝐷 ∪ {𝑋}) | |
14 | 13 | fneq2i 5218 | . . 3 ⊢ ((𝐹 ∪ {〈𝑋, 𝑌〉}) Fn 𝐸 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) |
15 | 12, 14 | bitri 183 | . 2 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) |
16 | 10, 15 | sylibr 133 | 1 ⊢ (𝜑 → 𝐺 Fn 𝐸) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 ∩ cin 3070 ∅c0 3363 {csn 3527 〈cop 3530 Fn wfn 5118 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-fun 5125 df-fn 5126 |
This theorem is referenced by: tfrlemisucfn 6221 tfr1onlemsucfn 6237 |
Copyright terms: Public domain | W3C validator |