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Mirrors > Home > MPE Home > Th. List > fsnunf | Structured version Visualization version GIF version |
Description: Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Ref | Expression |
---|---|
fsnunf | ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {〈𝑋, 𝑌〉}):(𝑆 ∪ {𝑋})⟶𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1054 | . . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝐹:𝑆⟶𝑇) | |
2 | simp2l 1080 | . . . . 5 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝑋 ∈ 𝑉) | |
3 | simp3 1056 | . . . . 5 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝑌 ∈ 𝑇) | |
4 | f1osng 6089 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑇) → {〈𝑋, 𝑌〉}:{𝑋}–1-1-onto→{𝑌}) | |
5 | 2, 3, 4 | syl2anc 691 | . . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → {〈𝑋, 𝑌〉}:{𝑋}–1-1-onto→{𝑌}) |
6 | f1of 6050 | . . . 4 ⊢ ({〈𝑋, 𝑌〉}:{𝑋}–1-1-onto→{𝑌} → {〈𝑋, 𝑌〉}:{𝑋}⟶{𝑌}) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → {〈𝑋, 𝑌〉}:{𝑋}⟶{𝑌}) |
8 | simp2r 1081 | . . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → ¬ 𝑋 ∈ 𝑆) | |
9 | disjsn 4192 | . . . 4 ⊢ ((𝑆 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝑆) | |
10 | 8, 9 | sylibr 223 | . . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝑆 ∩ {𝑋}) = ∅) |
11 | fun 5979 | . . 3 ⊢ (((𝐹:𝑆⟶𝑇 ∧ {〈𝑋, 𝑌〉}:{𝑋}⟶{𝑌}) ∧ (𝑆 ∩ {𝑋}) = ∅) → (𝐹 ∪ {〈𝑋, 𝑌〉}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌})) | |
12 | 1, 7, 10, 11 | syl21anc 1317 | . 2 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {〈𝑋, 𝑌〉}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌})) |
13 | snssi 4280 | . . . . 5 ⊢ (𝑌 ∈ 𝑇 → {𝑌} ⊆ 𝑇) | |
14 | 13 | 3ad2ant3 1077 | . . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → {𝑌} ⊆ 𝑇) |
15 | ssequn2 3748 | . . . 4 ⊢ ({𝑌} ⊆ 𝑇 ↔ (𝑇 ∪ {𝑌}) = 𝑇) | |
16 | 14, 15 | sylib 207 | . . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝑇 ∪ {𝑌}) = 𝑇) |
17 | 16 | feq3d 5945 | . 2 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → ((𝐹 ∪ {〈𝑋, 𝑌〉}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}) ↔ (𝐹 ∪ {〈𝑋, 𝑌〉}):(𝑆 ∪ {𝑋})⟶𝑇)) |
18 | 12, 17 | mpbid 221 | 1 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {〈𝑋, 𝑌〉}):(𝑆 ∪ {𝑋})⟶𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 〈cop 4131 ⟶wf 5800 –1-1-onto→wf1o 5803 |
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-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: fsnunf2 6357 fnchoice 38211 nnsum4primeseven 40216 nnsum4primesevenALTV 40217 |
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