Theorem f1omptsn 37905

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 = (𝑥 ∈ 𝐴 ↦ {𝑥})
f1omptsn.r	⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}

Assertion

Ref	Expression
f1omptsn	⊢ 𝐹:𝐴–1-1-onto→𝑅

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.

Step	Hyp	Ref	Expression
1		sneq 4604	. . . . . 6 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎})
2	1	cbvmptv 5219	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑎 ∈ 𝐴 ↦ {𝑎})
3	2	eqcomi 2778	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}) = (𝑥 ∈ 𝐴 ↦ {𝑥})
4		id 23	. . . . . . . 8 ⊢ (𝑢 = 𝑧 → 𝑢 = 𝑧)
5	4, 1	eqeqan12d 2783	. . . . . . 7 ⊢ ((𝑢 = 𝑧 ∧ 𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎}))
6	5	cbvrexdva 3252	. . . . . 6 ⊢ (𝑢 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}))
7	6	cbvabv 2839	. . . . 5 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}}
8	7	eqcomi 2778	. . . 4 ⊢ {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
9	3, 8	f1omptsnlem 37904	. . 3 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→{𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}}
10		f1omptsn.r	. . . . 5 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
11	10, 7	eqtri 2792	. . . 4 ⊢ 𝑅 = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}}
12		f1oeq3 6811	. . . 4 ⊢ (𝑅 = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} → ((𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅 ↔ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→{𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}}))
13	11, 12	ax-mp 5	. . 3 ⊢ ((𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅 ↔ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→{𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}})
14	9, 13	mpbir 234	. 2 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅
15		f1omptsn.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})
16	15, 2	eqtri 2792	. . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ {𝑎})
17		f1oeq1 6809	. . 3 ⊢ (𝐹 = (𝑎 ∈ 𝐴 ↦ {𝑎}) → (𝐹:𝐴–1-1-onto→𝑅 ↔ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅))
18	16, 17	ax-mp 5	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝑅 ↔ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅)
19	14, 18	mpbir 234	1 ⊢ 𝐹:𝐴–1-1-onto→𝑅

Syntax hints:   ↔ wb 209   = wceq 1567  {cab 2747  ∃wrex 3095  {csn 4594   ↦ cmpt 5196  –1-1-onto→wf1o 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is referenced by: (None)