Theorem f1ompt 7041

Description: Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)

Hypothesis

Ref	Expression
fmpt.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)

Assertion

Ref	Expression
f1ompt	⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑦,𝐹

Allowed substitution hints:   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem f1ompt

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 6651	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		dff1o4 6775	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
3	2	baib 536	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ◡𝐹 Fn 𝐵))
4	1, 3	syl 17	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ◡𝐹 Fn 𝐵))
5		fnres 6611	. . . . . 6 ⊢ ((◡𝐹 ↾ 𝐵) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑧 𝑦◡𝐹𝑧)
6		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
7		fmpt.1	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
8		nfmpt1 5200	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
9	7, 8	nfcxfr 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
10		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑦
11	6, 9, 10	nfbr 5139	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧𝐹𝑦
12		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)
13		breq1 5095	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧𝐹𝑦 ↔ 𝑥𝐹𝑦))
14		df-mpt 5176	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
15	7, 14	eqtri 2764	. . . . . . . . . . . 12 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
16	15	breqi 5098	. . . . . . . . . . 11 ⊢ (𝑥𝐹𝑦 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}𝑦)
17		df-br 5093	. . . . . . . . . . . 12 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)})
18		opabidw 5468	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
19	17, 18	bitri 274	. . . . . . . . . . 11 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
20	16, 19	bitri 274	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
21	13, 20	bitrdi 286	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)))
22	11, 12, 21	cbveuw 2605	. . . . . . . 8 ⊢ (∃!𝑧 𝑧𝐹𝑦 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
23		vex 3445	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
24		vex 3445	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
25	23, 24	brcnv 5824	. . . . . . . . 9 ⊢ (𝑦◡𝐹𝑧 ↔ 𝑧𝐹𝑦)
26	25	eubii 2583	. . . . . . . 8 ⊢ (∃!𝑧 𝑦◡𝐹𝑧 ↔ ∃!𝑧 𝑧𝐹𝑦)
27		df-reu 3350	. . . . . . . 8 ⊢ (∃!𝑥 ∈ 𝐴 𝑦 = 𝐶 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
28	22, 26, 27	3bitr4i 302	. . . . . . 7 ⊢ (∃!𝑧 𝑦◡𝐹𝑧 ↔ ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶)
29	28	ralbii 3092	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ∃!𝑧 𝑦◡𝐹𝑧 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶)
30	5, 29	bitri 274	. . . . 5 ⊢ ((◡𝐹 ↾ 𝐵) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶)
31		relcnv 6042	. . . . . . 7 ⊢ Rel ◡𝐹
32		df-rn 5631	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
33		frn 6658	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
34	32, 33	eqsstrrid 3981	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → dom ◡𝐹 ⊆ 𝐵)
35		relssres 5964	. . . . . . 7 ⊢ ((Rel ◡𝐹 ∧ dom ◡𝐹 ⊆ 𝐵) → (◡𝐹 ↾ 𝐵) = ◡𝐹)
36	31, 34, 35	sylancr 587	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 ↾ 𝐵) = ◡𝐹)
37	36	fneq1d 6578	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → ((◡𝐹 ↾ 𝐵) Fn 𝐵 ↔ ◡𝐹 Fn 𝐵))
38	30, 37	bitr3id 284	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶 ↔ ◡𝐹 Fn 𝐵))
39	4, 38	bitr4d 281	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶))
40	39	pm5.32i 575	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴–1-1-onto→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶))
41		f1of 6767	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
42	41	pm4.71ri 561	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴–1-1-onto→𝐵))
43	7	fmpt 7040	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
44	43	anbi1i 624	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶))
45	40, 42, 44	3bitr4i 302	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∃!weu 2566  ∀wral 3061  ∃!wreu 3347   ⊆ wss 3898  ⟨cop 4579   class class class wbr 5092  {copab 5154   ↦ cmpt 5175  ^◡ccnv 5619  dom cdm 5620  ran crn 5621   ↾ cres 5622  Rel wrel 5625   Fn wfn 6474  ⟶wf 6475  –1-1-onto→wf1o 6478

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486

This theorem is referenced by:  oaf1o  8465  xpf1o  9004  icoshftf1o  13307  fprodser  15758  dfod2  19267  gsummptf1o  19659  nbusgrf1o0  28025  cusgrfilem2  28112  numclwlk2lem2f1o  29031  f1mptrn  31257  xrmulc1cn  32178  poimirlem4  35894  poimirlem16  35906  poimirlem17  35907  poimirlem19  35909  poimirlem20  35910