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Theorem f1omptsn 33314
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 4220 . . . . . 6 (𝑥 = 𝑎 → {𝑥} = {𝑎})
21cbvmptv 4783 . . . . 5 (𝑥𝐴 ↦ {𝑥}) = (𝑎𝐴 ↦ {𝑎})
32eqcomi 2660 . . . 4 (𝑎𝐴 ↦ {𝑎}) = (𝑥𝐴 ↦ {𝑥})
4 id 22 . . . . . . . 8 (𝑢 = 𝑧𝑢 = 𝑧)
54, 1eqeqan12d 2667 . . . . . . 7 ((𝑢 = 𝑧𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎}))
65cbvrexdva 3208 . . . . . 6 (𝑢 = 𝑧 → (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑎𝐴 𝑧 = {𝑎}))
76cbvabv 2776 . . . . 5 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
87eqcomi 2660 . . . 4 {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
93, 8f1omptsnlem 33313 . . 3 (𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto→{𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
10 f1omptsn.r . . . . 5 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
1110, 7eqtri 2673 . . . 4 𝑅 = {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
12 f1oeq3 6167 . . . 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 2673 . . 3 𝐹 = (𝑎𝐴 ↦ {𝑎})
17 f1oeq1 6165 . . 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 1523  {cab 2637  wrex 2942  {csn 4210  cmpt 4762  1-1-ontowf1o 5925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by: (None)
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