Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 1132 | . . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝐹:𝑆⟶𝑇) | |
2 | simp2l 1195 | . . . . 5 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝑋 ∈ 𝑉) | |
3 | simp3 1134 | . . . . 5 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝑌 ∈ 𝑇) | |
4 | f1osng 6654 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑇) → {〈𝑋, 𝑌〉}:{𝑋}–1-1-onto→{𝑌}) | |
5 | 2, 3, 4 | syl2anc 586 | . . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → {〈𝑋, 𝑌〉}:{𝑋}–1-1-onto→{𝑌}) |
6 | f1of 6614 | . . . 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 1196 | . . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → ¬ 𝑋 ∈ 𝑆) | |
9 | disjsn 4646 | . . . 4 ⊢ ((𝑆 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝑆) | |
10 | 8, 9 | sylibr 236 | . . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝑆 ∩ {𝑋}) = ∅) |
11 | fun 6539 | . . 3 ⊢ (((𝐹:𝑆⟶𝑇 ∧ {〈𝑋, 𝑌〉}:{𝑋}⟶{𝑌}) ∧ (𝑆 ∩ {𝑋}) = ∅) → (𝐹 ∪ {〈𝑋, 𝑌〉}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌})) | |
12 | 1, 7, 10, 11 | syl21anc 835 | . 2 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {〈𝑋, 𝑌〉}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌})) |
13 | snssi 4740 | . . . . 5 ⊢ (𝑌 ∈ 𝑇 → {𝑌} ⊆ 𝑇) | |
14 | 13 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → {𝑌} ⊆ 𝑇) |
15 | ssequn2 4158 | . . . 4 ⊢ ({𝑌} ⊆ 𝑇 ↔ (𝑇 ∪ {𝑌}) = 𝑇) | |
16 | 14, 15 | sylib 220 | . . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝑇 ∪ {𝑌}) = 𝑇) |
17 | 16 | feq3d 6500 | . 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 234 | 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 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 {csn 4566 〈cop 4572 ⟶wf 6350 –1-1-onto→wf1o 6353 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 |
This theorem is referenced by: fsnunf2 6947 fnchoice 41284 nnsum4primeseven 43964 nnsum4primesevenALTV 43965 |
Copyright terms: Public domain | W3C validator |