Theorem f1omptsnlem 34619

Description: This is the core of the proof of f1omptsn 34620, 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 2823	. . . . . . 7 ⊢ {𝑥} = {𝑥}
3		snex 5334	. . . . . . . 8 ⊢ {𝑥} ∈ V
4		eqsbc3 3819	. . . . . . . 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 233	. . . . . 6 ⊢ [{𝑥} / 𝑢]𝑢 = {𝑥}
7		sbcel2 4369	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⦋{𝑥} / 𝑢⦌𝐴)
8		csbconstg 3904	. . . . . . . . . 10 ⊢ ({𝑥} ∈ V → ⦋{𝑥} / 𝑢⦌𝐴 = 𝐴)
9	3, 8	ax-mp 5	. . . . . . . . 9 ⊢ ⦋{𝑥} / 𝑢⦌𝐴 = 𝐴
10	9	eleq2i 2906	. . . . . . . 8 ⊢ (𝑥 ∈ ⦋{𝑥} / 𝑢⦌𝐴 ↔ 𝑥 ∈ 𝐴)
11	7, 10	bitri 277	. . . . . . 7 ⊢ ([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
12		f1omptsn.r	. . . . . . . . . . . . . 14 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
13	12	abeq2i 2950	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ 𝑅 ↔ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥})
14		df-rex 3146	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}))
15	13, 14	sylbbr 238	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝑅)
16	15	19.23bi 2190	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝑅)
17	16	sbcth 3789	. . . . . . . . . 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 3822	. . . . . . . . . 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 232	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢](𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) → [{𝑥} / 𝑢]𝑢 ∈ 𝑅)
22		sbcan 3823	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢](𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}) ↔ ([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}))
23		sbcel1v 3841	. . . . . . . 8 ⊢ ([{𝑥} / 𝑢]𝑢 ∈ 𝑅 ↔ {𝑥} ∈ 𝑅)
24	21, 22, 23	3imtr3i 293	. . . . . . 7 ⊢ (([{𝑥} / 𝑢]𝑥 ∈ 𝐴 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
25	11, 24	sylanbr 584	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
26	6, 25	mpan2 689	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝑅)
27	1, 26	fmpti 6878	. . . 4 ⊢ 𝐹:𝐴⟶𝑅
28	1	fvmpt2 6781	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ {𝑥} ∈ 𝑅) → (𝐹‘𝑥) = {𝑥})
29	26, 28	mpdan 685	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = {𝑥})
30		sneq 4579	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
31	30, 1, 3	fvmpt3i 6775	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = {𝑦})
32	29, 31	eqeqan12d 2840	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ {𝑥} = {𝑦}))
33		vex 3499	. . . . . . . 8 ⊢ 𝑥 ∈ V
34		sneqbg 4776	. . . . . . . 8 ⊢ (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
35	33, 34	ax-mp 5	. . . . . . 7 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
36	32, 35	syl6bb 289	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
37	36	biimpd 231	. . . . 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 7015	. . . 4 ⊢ (𝐹:𝐴–1-1→𝑅 ↔ (𝐹:𝐴⟶𝑅 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
40	27, 38, 39	mpbir2an 709	. . 3 ⊢ 𝐹:𝐴–1-1→𝑅
41		f1f1orn 6628	. . 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 34618	. . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
44	1	rneqi 5809	. . . 4 ⊢ ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ {𝑥})
45	43, 44, 12	3eqtr4i 2856	. . 3 ⊢ ran 𝐹 = 𝑅
46		f1oeq3 6608	. . 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 232	1 ⊢ 𝐹:𝐴–1-1-onto→𝑅

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2801  ∀wral 3140  ∃wrex 3141  Vcvv 3496  [wsbc 3774  ⦋csb 3885  {csn 4569   ↦ cmpt 5148  ran crn 5558  ⟶wf 6353  –1-1→wf1 6354  –1-1-onto→wf1o 6356  ‘cfv 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365

This theorem is referenced by:  f1omptsn  34620