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Mirrors > Home > MPE Home > Th. List > f1osng | Structured version Visualization version GIF version |
Description: A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.) |
Ref | Expression |
---|---|
f1osng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4639 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | 1 | f1oeq2d 6830 | . . 3 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} ↔ {⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏})) |
3 | opeq1 4874 | . . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩) | |
4 | 3 | sneqd 4641 | . . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩}) |
5 | 4 | f1oeq1d 6829 | . . 3 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏})) |
6 | 2, 5 | bitrd 279 | . 2 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏})) |
7 | sneq 4639 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵}) | |
8 | 7 | f1oeq3d 6831 | . . 3 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵})) |
9 | opeq2 4875 | . . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩) | |
10 | 9 | sneqd 4641 | . . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩}) |
11 | 10 | f1oeq1d 6829 | . . 3 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})) |
12 | 8, 11 | bitrd 279 | . 2 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})) |
13 | vex 3479 | . . 3 ⊢ 𝑎 ∈ V | |
14 | vex 3479 | . . 3 ⊢ 𝑏 ∈ V | |
15 | 13, 14 | f1osn 6874 | . 2 ⊢ {⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} |
16 | 6, 12, 15 | vtocl2g 3563 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4629 ⟨cop 4635 –1-1-onto→wf1o 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 |
This theorem is referenced by: f1sng 6876 f1oprswap 6878 f1oprg 6879 f1o2sn 7140 fsnunf 7183 fsnex 7281 suppsnop 8163 mapsnd 8880 ralxpmap 8890 en2sn 9041 en2snOLD 9042 enfixsn 9081 fseqenlem1 10019 canthp1lem2 10648 sumsnf 15689 prodsn 15906 prodsnf 15908 vdwlem8 16921 gsumws1 18719 symg1bas 19258 dprdsn 19906 eupthp1 29469 s1f1 32109 poimirlem16 36504 poimirlem17 36505 poimirlem19 36507 poimirlem20 36508 metakunt25 41009 mapfzcons 41454 sumsnd 43710 1hegrlfgr 46510 |
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