Theorem f1omptsn 36521

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 4637	. . . . . 6 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎})
2	1	cbvmptv 5260	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑎 ∈ 𝐴 ↦ {𝑎})
3	2	eqcomi 2739	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}) = (𝑥 ∈ 𝐴 ↦ {𝑥})
4		id 22	. . . . . . . 8 ⊢ (𝑢 = 𝑧 → 𝑢 = 𝑧)
5	4, 1	eqeqan12d 2744	. . . . . . 7 ⊢ ((𝑢 = 𝑧 ∧ 𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎}))
6	5	cbvrexdva 3235	. . . . . 6 ⊢ (𝑢 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}))
7	6	cbvabv 2803	. . . . 5 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}}
8	7	eqcomi 2739	. . . 4 ⊢ {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
9	3, 8	f1omptsnlem 36520	. . 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 2758	. . . 4 ⊢ 𝑅 = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}}
12		f1oeq3 6822	. . . 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 2758	. . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ {𝑎})
17		f1oeq1 6820	. . 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 1539  {cab 2707  ∃wrex 3068  {csn 4627   ↦ cmpt 5230  –1-1-onto→wf1o 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550

This theorem is referenced by: (None)