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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 5178 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {〈𝑋, 𝑌〉} Fn {𝑋}) | |
5 | 2, 3, 4 | syl2anc 409 | . . 3 ⊢ (𝜑 → {〈𝑋, 𝑌〉} Fn {𝑋}) |
6 | fnunop.d | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷) | |
7 | disjsn 3593 | . . . 4 ⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐷) | |
8 | 6, 7 | sylibr 133 | . . 3 ⊢ (𝜑 → (𝐷 ∩ {𝑋}) = ∅) |
9 | fnun 5237 | . . 3 ⊢ (((𝐹 Fn 𝐷 ∧ {〈𝑋, 𝑌〉} Fn {𝑋}) ∧ (𝐷 ∩ {𝑋}) = ∅) → (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) | |
10 | 1, 5, 8, 9 | syl21anc 1216 | . 2 ⊢ (𝜑 → (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) |
11 | fnunop.g | . . . 4 ⊢ 𝐺 = (𝐹 ∪ {〈𝑋, 𝑌〉}) | |
12 | 11 | fneq1i 5225 | . . 3 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn 𝐸) |
13 | fnunop.e | . . . 4 ⊢ 𝐸 = (𝐷 ∪ {𝑋}) | |
14 | 13 | fneq2i 5226 | . . 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 1332 ∈ wcel 1481 Vcvv 2689 ∪ cun 3074 ∩ cin 3075 ∅c0 3368 {csn 3532 〈cop 3535 Fn wfn 5126 |
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-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 |
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-ral 2422 df-rex 2423 df-v 2691 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-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-fun 5133 df-fn 5134 |
This theorem is referenced by: tfrlemisucfn 6229 tfr1onlemsucfn 6245 |
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