Theorem f1omptsnlem 32359

Description: This is the core of the proof of f1omptsn 32360, 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 2610	. . . . . . 7 ⊢ {𝑥} = {𝑥}
3		snex 4835	. . . . . . . 8 ⊢ {𝑥} ∈ V
4		eqsbc3 3442	. . . . . . . 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 220	. . . . . 6 ⊢ [{𝑥} / 𝑢]𝑢 = {𝑥}
7		sbcel2 3941	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⦋{𝑥} / 𝑢⦌𝐴)
8		csbconstg 3512	. . . . . . . . . 10 ⊢ ({𝑥} ∈ V → ⦋{𝑥} / 𝑢⦌𝐴 = 𝐴)
9	3, 8	ax-mp 5	. . . . . . . . 9 ⊢ ⦋{𝑥} / 𝑢⦌𝐴 = 𝐴
10	9	eleq2i 2680	. . . . . . . 8 ⊢ (𝑥 ∈ ⦋{𝑥} / 𝑢⦌𝐴 ↔ 𝑥 ∈ 𝐴)
11	7, 10	bitri 263	. . . . . . 7 ⊢ ([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
12		f1omptsn.r	. . . . . . . . . . . . . 14 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
13	12	abeq2i 2722	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ 𝑅 ↔ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥})
14		df-rex 2902	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}))
15	13, 14	sylbbr 225	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝑅)
16	15	19.23bi 2049	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝑅)
17	16	sbcth 3417	. . . . . . . . . 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 3444	. . . . . . . . . 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 219	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢](𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → [{𝑥} / 𝑢]𝑢 ∈ 𝑅)
22		sbcan 3445	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢](𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) ↔ ([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}))
23		sbcel1v 3462	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢]𝑢 ∈ 𝑅 ↔ {𝑥} ∈ 𝑅)
24	21, 22, 23	3imtr3i 279	. . . . . . 7 ⊢ (([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
25	11, 24	sylanbr 489	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
26	6, 25	mpan2 703	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝑅)
27	1, 26	fmpti 6291	. . . 4 ⊢ 𝐹:𝐴⟶𝑅
28	1	fvmpt2 6200	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ {𝑥} ∈ 𝑅) → (𝐹‘𝑥) = {𝑥})
29	26, 28	mpdan 699	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = {𝑥})
30		sneq 4135	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
31	30, 1, 3	fvmpt3i 6196	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = {𝑦})
32	29, 31	eqeqan12d 2626	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ {𝑥} = {𝑦}))
33		vex 3176	. . . . . . . 8 ⊢ 𝑥 ∈ V
34		sneqbg 4314	. . . . . . . 8 ⊢ (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
35	33, 34	ax-mp 5	. . . . . . 7 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
36	32, 35	syl6bb 275	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
37	36	biimpd 218	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
38	37	rgen2a 2960	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
39		dff13 6416	. . . 4 ⊢ (𝐹:𝐴–1-1→𝑅 ↔ (𝐹:𝐴⟶𝑅 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
40	27, 38, 39	mpbir2an 957	. . 3 ⊢ 𝐹:𝐴–1-1→𝑅
41		f1f1orn 6061	. . 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 32358	. . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
44	1	rneqi 5273	. . . 4 ⊢ ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ {𝑥})
45	43, 44, 12	3eqtr4i 2642	. . 3 ⊢ ran 𝐹 = 𝑅
46		f1oeq3 6042	. . 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 219	1 ⊢ 𝐹:𝐴–1-1-onto→𝑅

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173  [wsbc 3402  ⦋csb 3499  {csn 4125   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812

This theorem is referenced by:  f1omptsn  32360