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Theorem f1omptsn 32813
Description: A function mapping to singletons is bijective onto a set of singletons. (Contributed by ML, 16-Jul-2020.)
Hypotheses
Ref Expression
f1omptsn.f 𝐹 = (𝑥𝐴 ↦ {𝑥})
f1omptsn.r 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
Assertion
Ref Expression
f1omptsn 𝐹:𝐴1-1-onto𝑅
Distinct variable group:   𝑢,𝐴,𝑥
Allowed substitution hints:   𝑅(𝑥,𝑢)   𝐹(𝑥,𝑢)

Proof of Theorem f1omptsn
Dummy variables 𝑎 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sneq 4158 . . . . . 6 (𝑥 = 𝑎 → {𝑥} = {𝑎})
21cbvmptv 4710 . . . . 5 (𝑥𝐴 ↦ {𝑥}) = (𝑎𝐴 ↦ {𝑎})
32eqcomi 2630 . . . 4 (𝑎𝐴 ↦ {𝑎}) = (𝑥𝐴 ↦ {𝑥})
4 id 22 . . . . . . . 8 (𝑢 = 𝑧𝑢 = 𝑧)
54, 1eqeqan12d 2637 . . . . . . 7 ((𝑢 = 𝑧𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎}))
65cbvrexdva 3166 . . . . . 6 (𝑢 = 𝑧 → (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑎𝐴 𝑧 = {𝑎}))
76cbvabv 2744 . . . . 5 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
87eqcomi 2630 . . . 4 {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
93, 8f1omptsnlem 32812 . . 3 (𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto→{𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
10 f1omptsn.r . . . . 5 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
1110, 7eqtri 2643 . . . 4 𝑅 = {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
12 f1oeq3 6086 . . . 4 (𝑅 = {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}} → ((𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto𝑅 ↔ (𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto→{𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}))
1311, 12ax-mp 5 . . 3 ((𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto𝑅 ↔ (𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto→{𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}})
149, 13mpbir 221 . 2 (𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto𝑅
15 f1omptsn.f . . . 4 𝐹 = (𝑥𝐴 ↦ {𝑥})
1615, 2eqtri 2643 . . 3 𝐹 = (𝑎𝐴 ↦ {𝑎})
17 f1oeq1 6084 . . 3 (𝐹 = (𝑎𝐴 ↦ {𝑎}) → (𝐹:𝐴1-1-onto𝑅 ↔ (𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto𝑅))
1816, 17ax-mp 5 . 2 (𝐹:𝐴1-1-onto𝑅 ↔ (𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto𝑅)
1914, 18mpbir 221 1 𝐹:𝐴1-1-onto𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  {cab 2607  wrex 2908  {csn 4148  cmpt 4673  1-1-ontowf1o 5846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855
This theorem is referenced by: (None)
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