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Theorem f1omptsn 35271
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 4565 . . . . . 6 (𝑥 = 𝑎 → {𝑥} = {𝑎})
21cbvmptv 5172 . . . . 5 (𝑥𝐴 ↦ {𝑥}) = (𝑎𝐴 ↦ {𝑎})
32eqcomi 2747 . . . 4 (𝑎𝐴 ↦ {𝑎}) = (𝑥𝐴 ↦ {𝑥})
4 id 22 . . . . . . . 8 (𝑢 = 𝑧𝑢 = 𝑧)
54, 1eqeqan12d 2752 . . . . . . 7 ((𝑢 = 𝑧𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎}))
65cbvrexdva 3382 . . . . . 6 (𝑢 = 𝑧 → (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑎𝐴 𝑧 = {𝑎}))
76cbvabv 2812 . . . . 5 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
87eqcomi 2747 . . . 4 {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
93, 8f1omptsnlem 35270 . . 3 (𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto→{𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
10 f1omptsn.r . . . . 5 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
1110, 7eqtri 2766 . . . 4 𝑅 = {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
12 f1oeq3 6669 . . . 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 234 . 2 (𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto𝑅
15 f1omptsn.f . . . 4 𝐹 = (𝑥𝐴 ↦ {𝑥})
1615, 2eqtri 2766 . . 3 𝐹 = (𝑎𝐴 ↦ {𝑎})
17 f1oeq1 6667 . . 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 234 1 𝐹:𝐴1-1-onto𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1543  {cab 2715  wrex 3063  {csn 4555  cmpt 5149  1-1-ontowf1o 6396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-sbc 3709  df-csb 3826  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-mpt 5150  df-id 5469  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578  df-iota 6355  df-fun 6399  df-fn 6400  df-f 6401  df-f1 6402  df-fo 6403  df-f1o 6404  df-fv 6405
This theorem is referenced by: (None)
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