Theorem f1omptsnlem 37338

Description: This is the core of the proof of f1omptsn 37339, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 15-Jul-2020.)

Hypotheses

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

Assertion

Ref	Expression
f1omptsnlem	⊢ 𝐹:𝐴–1-1-onto→𝑅

Distinct variable groups:   𝑥,𝐴,𝑢   𝑥,𝐹   𝑢,𝑅,𝑥

Allowed substitution hint:   𝐹(𝑢)

Proof of Theorem f1omptsnlem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1omptsn.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})
2		eqid 2736	. . . . . . 7 ⊢ {𝑥} = {𝑥}
3		vsnex 5433	. . . . . . . 8 ⊢ {𝑥} ∈ V
4		eqsbc1 3834	. . . . . . . 8 ⊢ ({𝑥} ∈ V → ([{𝑥} / 𝑢]𝑢 = {𝑥} ↔ {𝑥} = {𝑥}))
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ ([{𝑥} / 𝑢]𝑢 = {𝑥} ↔ {𝑥} = {𝑥})
6	2, 5	mpbir 231	. . . . . 6 ⊢ [{𝑥} / 𝑢]𝑢 = {𝑥}
7		sbcel2 4417	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⦋{𝑥} / 𝑢⦌𝐴)
8		csbconstg 3917	. . . . . . . . . 10 ⊢ ({𝑥} ∈ V → ⦋{𝑥} / 𝑢⦌𝐴 = 𝐴)
9	3, 8	ax-mp 5	. . . . . . . . 9 ⊢ ⦋{𝑥} / 𝑢⦌𝐴 = 𝐴
10	9	eleq2i 2832	. . . . . . . 8 ⊢ (𝑥 ∈ ⦋{𝑥} / 𝑢⦌𝐴 ↔ 𝑥 ∈ 𝐴)
11	7, 10	bitri 275	. . . . . . 7 ⊢ ([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
12		f1omptsn.r	. . . . . . . . . . . . . 14 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
13	12	eqabri 2884	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ 𝑅 ↔ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥})
14		df-rex 3070	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}))
15	13, 14	sylbbr 236	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝑅)
16	15	19.23bi 2190	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝑅)
17	16	sbcth 3802	. . . . . . . . . 10 ⊢ ({𝑥} ∈ V → [{𝑥} / 𝑢]((𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝑅))
18	3, 17	ax-mp 5	. . . . . . . . 9 ⊢ [{𝑥} / 𝑢]((𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝑅)
19		sbcimg 3836	. . . . . . . . . 10 ⊢ ({𝑥} ∈ V → ([{𝑥} / 𝑢]((𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝑅) ↔ ([{𝑥} / 𝑢](𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → [{𝑥} / 𝑢]𝑢 ∈ 𝑅)))
20	3, 19	ax-mp 5	. . . . . . . . 9 ⊢ ([{𝑥} / 𝑢]((𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝑅) ↔ ([{𝑥} / 𝑢](𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → [{𝑥} / 𝑢]𝑢 ∈ 𝑅))
21	18, 20	mpbi 230	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢](𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → [{𝑥} / 𝑢]𝑢 ∈ 𝑅)
22		sbcan 3837	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢](𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) ↔ ([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}))
23		sbcel1v 3855	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢]𝑢 ∈ 𝑅 ↔ {𝑥} ∈ 𝑅)
24	21, 22, 23	3imtr3i 291	. . . . . . 7 ⊢ (([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
25	11, 24	sylanbr 582	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
26	6, 25	mpan2 691	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝑅)
27	1, 26	fmpti 7131	. . . 4 ⊢ 𝐹:𝐴⟶𝑅
28	1	fvmpt2 7026	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ {𝑥} ∈ 𝑅) → (𝐹‘𝑥) = {𝑥})
29	26, 28	mpdan 687	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = {𝑥})
30		sneq 4635	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
31	30, 1, 3	fvmpt3i 7020	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = {𝑦})
32	29, 31	eqeqan12d 2750	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ {𝑥} = {𝑦}))
33		vex 3483	. . . . . . . 8 ⊢ 𝑥 ∈ V
34		sneqbg 4842	. . . . . . . 8 ⊢ (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
35	33, 34	ax-mp 5	. . . . . . 7 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
36	32, 35	bitrdi 287	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
37	36	biimpd 229	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
38	37	rgen2 3198	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
39		dff13 7276	. . . 4 ⊢ (𝐹:𝐴–1-1→𝑅 ↔ (𝐹:𝐴⟶𝑅 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
40	27, 38, 39	mpbir2an 711	. . 3 ⊢ 𝐹:𝐴–1-1→𝑅
41		f1f1orn 6858	. . 3 ⊢ (𝐹:𝐴–1-1→𝑅 → 𝐹:𝐴–1-1-onto→ran 𝐹)
42	40, 41	ax-mp 5	. 2 ⊢ 𝐹:𝐴–1-1-onto→ran 𝐹
43		rnmptsn 37337	. . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
44	1	rneqi 5947	. . . 4 ⊢ ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ {𝑥})
45	43, 44, 12	3eqtr4i 2774	. . 3 ⊢ ran 𝐹 = 𝑅
46		f1oeq3 6837	. . 3 ⊢ (ran 𝐹 = 𝑅 → (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ 𝐹:𝐴–1-1-onto→𝑅))
47	45, 46	ax-mp 5	. 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ 𝐹:𝐴–1-1-onto→𝑅)
48	42, 47	mpbi 230	1 ⊢ 𝐹:𝐴–1-1-onto→𝑅

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2713  ∀wral 3060  ∃wrex 3069  Vcvv 3479  [wsbc 3787  ⦋csb 3898  {csn 4625   ↦ cmpt 5224  ran crn 5685  ⟶wf 6556  –1-1→wf1 6557  –1-1-onto→wf1o 6559  ‘cfv 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568

This theorem is referenced by:  f1omptsn  37339