Theorem f1omptsn 37056

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 4635	. . . . . 6 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎})
2	1	cbvmptv 5258	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑎 ∈ 𝐴 ↦ {𝑎})
3	2	eqcomi 2735	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}) = (𝑥 ∈ 𝐴 ↦ {𝑥})
4		id 22	. . . . . . . 8 ⊢ (𝑢 = 𝑧 → 𝑢 = 𝑧)
5	4, 1	eqeqan12d 2740	. . . . . . 7 ⊢ ((𝑢 = 𝑧 ∧ 𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎}))
6	5	cbvrexdva 3228	. . . . . 6 ⊢ (𝑢 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}))
7	6	cbvabv 2799	. . . . 5 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}}
8	7	eqcomi 2735	. . . 4 ⊢ {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
9	3, 8	f1omptsnlem 37055	. . 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 2754	. . . 4 ⊢ 𝑅 = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}}
12		f1oeq3 6824	. . . 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 230	. 2 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅
15		f1omptsn.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})
16	15, 2	eqtri 2754	. . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ {𝑎})
17		f1oeq1 6822	. . 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 230	1 ⊢ 𝐹:𝐴–1-1-onto→𝑅

Syntax hints:   ↔ wb 205   = wceq 1534  {cab 2703  ∃wrex 3060  {csn 4625   ↦ cmpt 5228  –1-1-onto→wf1o 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4325  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553

This theorem is referenced by: (None)