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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 6399 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {〈𝑋, 𝑌〉} Fn {𝑋}) | |
5 | 2, 3, 4 | syl2anc 584 | . . 3 ⊢ (𝜑 → {〈𝑋, 𝑌〉} Fn {𝑋}) |
6 | fnunop.d | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷) | |
7 | disjsn 4639 | . . . 4 ⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐷) | |
8 | 6, 7 | sylibr 235 | . . 3 ⊢ (𝜑 → (𝐷 ∩ {𝑋}) = ∅) |
9 | fnun 6456 | . . 3 ⊢ (((𝐹 Fn 𝐷 ∧ {〈𝑋, 𝑌〉} Fn {𝑋}) ∧ (𝐷 ∩ {𝑋}) = ∅) → (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) | |
10 | 1, 5, 8, 9 | syl21anc 833 | . 2 ⊢ (𝜑 → (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) |
11 | fnunop.g | . . . 4 ⊢ 𝐺 = (𝐹 ∪ {〈𝑋, 𝑌〉}) | |
12 | 11 | fneq1i 6443 | . . 3 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn 𝐸) |
13 | fnunop.e | . . . 4 ⊢ 𝐸 = (𝐷 ∪ {𝑋}) | |
14 | 13 | fneq2i 6444 | . . 3 ⊢ ((𝐹 ∪ {〈𝑋, 𝑌〉}) Fn 𝐸 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) |
15 | 12, 14 | bitri 276 | . 2 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}) Fn (𝐷 ∪ {𝑋})) |
16 | 10, 15 | sylibr 235 | 1 ⊢ (𝜑 → 𝐺 Fn 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ∩ cin 3932 ∅c0 4288 {csn 4557 〈cop 4563 Fn wfn 6343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-fun 6350 df-fn 6351 |
This theorem is referenced by: (None) |
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