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Mirrors > Home > MPE Home > Th. List > fsnunf2 | Structured version Visualization version GIF version |
Description: Adjoining a point to a punctured function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Ref | Expression |
---|---|
fsnunf2 | ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {〈𝑋, 𝑌〉}):𝑆⟶𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . . 3 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → 𝐹:(𝑆 ∖ {𝑋})⟶𝑇) | |
2 | simp2 1136 | . . 3 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → 𝑋 ∈ 𝑆) | |
3 | neldifsnd 4798 | . . 3 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) | |
4 | simp3 1137 | . . 3 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → 𝑌 ∈ 𝑇) | |
5 | fsnunf 7205 | . . 3 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ (𝑋 ∈ 𝑆 ∧ ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {〈𝑋, 𝑌〉}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇) | |
6 | 1, 2, 3, 4, 5 | syl121anc 1374 | . 2 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {〈𝑋, 𝑌〉}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇) |
7 | difsnid 4815 | . . . 4 ⊢ (𝑋 ∈ 𝑆 → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆) | |
8 | 7 | 3ad2ant2 1133 | . . 3 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆) |
9 | 8 | feq2d 6723 | . 2 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → ((𝐹 ∪ {〈𝑋, 𝑌〉}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇 ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}):𝑆⟶𝑇)) |
10 | 6, 9 | mpbid 232 | 1 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {〈𝑋, 𝑌〉}):𝑆⟶𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ∪ cun 3961 {csn 4631 〈cop 4637 ⟶wf 6559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 |
This theorem is referenced by: fsets 17203 islindf4 21876 |
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