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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 1137 | . . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝐹:𝑆⟶𝑇) | |
2 | simp2l 1200 | . . . . 5 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝑋 ∈ 𝑉) | |
3 | simp3 1139 | . . . . 5 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝑌 ∈ 𝑇) | |
4 | f1osng 6671 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑇) → {〈𝑋, 𝑌〉}:{𝑋}–1-1-onto→{𝑌}) | |
5 | 2, 3, 4 | syl2anc 587 | . . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → {〈𝑋, 𝑌〉}:{𝑋}–1-1-onto→{𝑌}) |
6 | f1of 6631 | . . . 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 1201 | . . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → ¬ 𝑋 ∈ 𝑆) | |
9 | disjsn 4612 | . . . 4 ⊢ ((𝑆 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝑆) | |
10 | 8, 9 | sylibr 237 | . . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝑆 ∩ {𝑋}) = ∅) |
11 | fun 6551 | . . 3 ⊢ (((𝐹:𝑆⟶𝑇 ∧ {〈𝑋, 𝑌〉}:{𝑋}⟶{𝑌}) ∧ (𝑆 ∩ {𝑋}) = ∅) → (𝐹 ∪ {〈𝑋, 𝑌〉}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌})) | |
12 | 1, 7, 10, 11 | syl21anc 837 | . 2 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {〈𝑋, 𝑌〉}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌})) |
13 | snssi 4706 | . . . . 5 ⊢ (𝑌 ∈ 𝑇 → {𝑌} ⊆ 𝑇) | |
14 | 13 | 3ad2ant3 1136 | . . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → {𝑌} ⊆ 𝑇) |
15 | ssequn2 4083 | . . . 4 ⊢ ({𝑌} ⊆ 𝑇 ↔ (𝑇 ∪ {𝑌}) = 𝑇) | |
16 | 14, 15 | sylib 221 | . . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝑇 ∪ {𝑌}) = 𝑇) |
17 | 16 | feq3d 6502 | . 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 235 | 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∪ cun 3851 ∩ cin 3852 ⊆ wss 3853 ∅c0 4221 {csn 4526 〈cop 4532 ⟶wf 6346 –1-1-onto→wf1o 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-br 5041 df-opab 5103 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 |
This theorem is referenced by: fsnunf2 6971 fnchoice 42151 nnsum4primeseven 44834 nnsum4primesevenALTV 44835 |
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