Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 1054 | . . 3 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → 𝐹:(𝑆 ∖ {𝑋})⟶𝑇) | |
2 | simp2 1055 | . . 3 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → 𝑋 ∈ 𝑆) | |
3 | neldifsnd 4263 | . . 3 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) | |
4 | simp3 1056 | . . 3 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → 𝑌 ∈ 𝑇) | |
5 | fsnunf 6356 | . . 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 1323 | . 2 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {〈𝑋, 𝑌〉}):((𝑆 ∖ {𝑋}) ∪ {𝑋})⟶𝑇) |
7 | difsnid 4282 | . . . 4 ⊢ (𝑋 ∈ 𝑆 → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆) | |
8 | 7 | 3ad2ant2 1076 | . . 3 ⊢ ((𝐹:(𝑆 ∖ {𝑋})⟶𝑇 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑇) → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆) |
9 | 8 | feq2d 5944 | . 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 221 | 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 1031 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 {csn 4125 〈cop 4131 ⟶wf 5800 |
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-ne 2782 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-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 |
This theorem is referenced by: fsets 15723 islindf4 19996 |
Copyright terms: Public domain | W3C validator |