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Theorem f1omptsn 33495
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 4331 . . . . . 6 (𝑥 = 𝑎 → {𝑥} = {𝑎})
21cbvmptv 4902 . . . . 5 (𝑥𝐴 ↦ {𝑥}) = (𝑎𝐴 ↦ {𝑎})
32eqcomi 2769 . . . 4 (𝑎𝐴 ↦ {𝑎}) = (𝑥𝐴 ↦ {𝑥})
4 id 22 . . . . . . . 8 (𝑢 = 𝑧𝑢 = 𝑧)
54, 1eqeqan12d 2776 . . . . . . 7 ((𝑢 = 𝑧𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎}))
65cbvrexdva 3317 . . . . . 6 (𝑢 = 𝑧 → (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑎𝐴 𝑧 = {𝑎}))
76cbvabv 2885 . . . . 5 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
87eqcomi 2769 . . . 4 {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
93, 8f1omptsnlem 33494 . . 3 (𝑎𝐴 ↦ {𝑎}):𝐴1-1-onto→{𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
10 f1omptsn.r . . . . 5 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
1110, 7eqtri 2782 . . . 4 𝑅 = {𝑧 ∣ ∃𝑎𝐴 𝑧 = {𝑎}}
12 f1oeq3 6290 . . . 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 2782 . . 3 𝐹 = (𝑎𝐴 ↦ {𝑎})
17 f1oeq1 6288 . . 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 1632  {cab 2746  wrex 3051  {csn 4321  cmpt 4881  1-1-ontowf1o 6048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057
This theorem is referenced by: (None)
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