Theorem f1omptsn 37318

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 4595	. . . . . 6 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎})
2	1	cbvmptv 5206	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑎 ∈ 𝐴 ↦ {𝑎})
3	2	eqcomi 2738	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}) = (𝑥 ∈ 𝐴 ↦ {𝑥})
4		id 22	. . . . . . . 8 ⊢ (𝑢 = 𝑧 → 𝑢 = 𝑧)
5	4, 1	eqeqan12d 2743	. . . . . . 7 ⊢ ((𝑢 = 𝑧 ∧ 𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎}))
6	5	cbvrexdva 3216	. . . . . 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 2738	. . . 4 ⊢ {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
9	3, 8	f1omptsnlem 37317	. . 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 2752	. . . 4 ⊢ 𝑅 = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}}
12		f1oeq3 6772	. . . 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 231	. 2 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅
15		f1omptsn.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})
16	15, 2	eqtri 2752	. . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ {𝑎})
17		f1oeq1 6770	. . 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 231	1 ⊢ 𝐹:𝐴–1-1-onto→𝑅

Syntax hints:   ↔ wb 206   = wceq 1540  {cab 2707  ∃wrex 3053  {csn 4585   ↦ cmpt 5183  –1-1-onto→wf1o 6498

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507

This theorem is referenced by: (None)