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Theorem f1omptsn 35854
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 4597 . . . . . 6 (𝑥 = 𝑎 → {𝑥} = {𝑎})
21cbvmptv 5219 . . . . 5 (𝑥𝐴 ↦ {𝑥}) = (𝑎𝐴 ↦ {𝑎})
32eqcomi 2742 . . . 4 (𝑎𝐴 ↦ {𝑎}) = (𝑥𝐴 ↦ {𝑥})
4 id 22 . . . . . . . 8 (𝑢 = 𝑧𝑢 = 𝑧)
54, 1eqeqan12d 2747 . . . . . . 7 ((𝑢 = 𝑧𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎}))
65cbvrexdva 3326 . . . . . 6 (𝑢 = 𝑧 → (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑎𝐴 𝑧 = {𝑎}))
76cbvabv 2806 . . . . 5 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
87eqcomi 2742 . . . 4 {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
93, 8f1omptsnlem 35853 . . 3 (𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto→{𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
10 f1omptsn.r . . . . 5 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
1110, 7eqtri 2761 . . . 4 𝑅 = {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
12 f1oeq3 6775 . . . 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 230 . 2 (𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto𝑅
15 f1omptsn.f . . . 4 𝐹 = (𝑥𝐴 ↦ {𝑥})
1615, 2eqtri 2761 . . 3 𝐹 = (𝑎𝐴 ↦ {𝑎})
17 f1oeq1 6773 . . 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 230 1 𝐹:𝐴1-1-onto𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  {cab 2710  wrex 3070  {csn 4587  cmpt 5189  1-1-ontowf1o 6496
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505
This theorem is referenced by: (None)
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