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Mirrors > Home > MPE Home > Th. List > fnunsn | Structured version Visualization version 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 5852 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {〈𝑋, 𝑌〉} Fn {𝑋}) | |
5 | 2, 3, 4 | syl2anc 691 | . . 3 ⊢ (𝜑 → {〈𝑋, 𝑌〉} Fn {𝑋}) |
6 | fnunop.d | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷) | |
7 | disjsn 4192 | . . . 4 ⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐷) | |
8 | 6, 7 | sylibr 223 | . . 3 ⊢ (𝜑 → (𝐷 ∩ {𝑋}) = ∅) |
9 | fnun 5911 | . . 3 ⊢ (((𝐹 Fn 𝐷 ∧ {〈𝑋, 𝑌〉} Fn {𝑋}) ∧ (𝐷 ∩ {𝑋}) = ∅) → (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) | |
10 | 1, 5, 8, 9 | syl21anc 1317 | . 2 ⊢ (𝜑 → (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) |
11 | fnunop.g | . . . 4 ⊢ 𝐺 = (𝐹 ∪ {〈𝑋, 𝑌〉}) | |
12 | 11 | fneq1i 5899 | . . 3 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn 𝐸) |
13 | fnunop.e | . . . 4 ⊢ 𝐸 = (𝐷 ∪ {𝑋}) | |
14 | 13 | fneq2i 5900 | . . 3 ⊢ ((𝐹 ∪ {〈𝑋, 𝑌〉}) Fn 𝐸 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) |
15 | 12, 14 | bitri 263 | . 2 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) |
16 | 10, 15 | sylibr 223 | 1 ⊢ (𝜑 → 𝐺 Fn 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {csn 4125 〈cop 4131 Fn wfn 5799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-fun 5806 df-fn 5807 |
This theorem is referenced by: (None) |
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