Theorem f1omptsnlem 37302

Description: This is the core of the proof of f1omptsn 37303, 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 2740	. . . . . . 7 ⊢ {𝑥} = {𝑥}
3		vsnex 5449	. . . . . . . 8 ⊢ {𝑥} ∈ V
4		eqsbc1 3854	. . . . . . . 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 4441	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⦋{𝑥} / 𝑢⦌𝐴)
8		csbconstg 3940	. . . . . . . . . 10 ⊢ ({𝑥} ∈ V → ⦋{𝑥} / 𝑢⦌𝐴 = 𝐴)
9	3, 8	ax-mp 5	. . . . . . . . 9 ⊢ ⦋{𝑥} / 𝑢⦌𝐴 = 𝐴
10	9	eleq2i 2836	. . . . . . . 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 2888	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ 𝑅 ↔ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥})
14		df-rex 3077	. . . . . . . . . . . . 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 2192	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝑅)
17	16	sbcth 3819	. . . . . . . . . 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 3856	. . . . . . . . . 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 3857	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢](𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) ↔ ([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}))
23		sbcel1v 3875	. . . . . . . 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 581	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
26	6, 25	mpan2 690	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝑅)
27	1, 26	fmpti 7146	. . . 4 ⊢ 𝐹:𝐴⟶𝑅
28	1	fvmpt2 7040	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ {𝑥} ∈ 𝑅) → (𝐹‘𝑥) = {𝑥})
29	26, 28	mpdan 686	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = {𝑥})
30		sneq 4658	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
31	30, 1, 3	fvmpt3i 7034	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = {𝑦})
32	29, 31	eqeqan12d 2754	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ {𝑥} = {𝑦}))
33		vex 3492	. . . . . . . 8 ⊢ 𝑥 ∈ V
34		sneqbg 4868	. . . . . . . 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 3205	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
39		dff13 7292	. . . 4 ⊢ (𝐹:𝐴–1-1→𝑅 ↔ (𝐹:𝐴⟶𝑅 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
40	27, 38, 39	mpbir2an 710	. . 3 ⊢ 𝐹:𝐴–1-1→𝑅
41		f1f1orn 6873	. . 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 37301	. . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
44	1	rneqi 5962	. . . 4 ⊢ ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ {𝑥})
45	43, 44, 12	3eqtr4i 2778	. . 3 ⊢ ran 𝐹 = 𝑅
46		f1oeq3 6852	. . 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 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488  [wsbc 3804  ⦋csb 3921  {csn 4648   ↦ cmpt 5249  ran crn 5701  ⟶wf 6569  –1-1→wf1 6570  –1-1-onto→wf1o 6572  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by:  f1omptsn  37303