Theorem f1ompt 7065

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 6670	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		dff1o4 6790	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
3	2	baib 535	. . . . 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 6627	. . . . . 6 ⊢ ((◡𝐹 ↾ 𝐵) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑧 𝑦◡𝐹𝑧)
6		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
7		fmpt.1	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
8		nfmpt1 5199	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
9	7, 8	nfcxfr 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
10		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑦
11	6, 9, 10	nfbr 5147	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧𝐹𝑦
12		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)
13		breq1 5103	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧𝐹𝑦 ↔ 𝑥𝐹𝑦))
14		df-mpt 5182	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
15	7, 14	eqtri 2760	. . . . . . . . . . . 12 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
16	15	breqi 5106	. . . . . . . . . . 11 ⊢ (𝑥𝐹𝑦 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}𝑦)
17		df-br 5101	. . . . . . . . . . . 12 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)})
18		opabidw 5480	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
19	17, 18	bitri 275	. . . . . . . . . . 11 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
20	16, 19	bitri 275	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
21	13, 20	bitrdi 287	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)))
22	11, 12, 21	cbveuw 2607	. . . . . . . 8 ⊢ (∃!𝑧 𝑧𝐹𝑦 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
23		vex 3446	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
24		vex 3446	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
25	23, 24	brcnv 5839	. . . . . . . . 9 ⊢ (𝑦◡𝐹𝑧 ↔ 𝑧𝐹𝑦)
26	25	eubii 2586	. . . . . . . 8 ⊢ (∃!𝑧 𝑦◡𝐹𝑧 ↔ ∃!𝑧 𝑧𝐹𝑦)
27		df-reu 3353	. . . . . . . 8 ⊢ (∃!𝑥 ∈ 𝐴 𝑦 = 𝐶 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
28	22, 26, 27	3bitr4i 303	. . . . . . 7 ⊢ (∃!𝑧 𝑦◡𝐹𝑧 ↔ ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶)
29	28	ralbii 3084	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ∃!𝑧 𝑦◡𝐹𝑧 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶)
30	5, 29	bitri 275	. . . . 5 ⊢ ((◡𝐹 ↾ 𝐵) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶)
31		relcnv 6071	. . . . . . 7 ⊢ Rel ◡𝐹
32		df-rn 5643	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
33		frn 6677	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
34	32, 33	eqsstrrid 3975	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → dom ◡𝐹 ⊆ 𝐵)
35		relssres 5989	. . . . . . 7 ⊢ ((Rel ◡𝐹 ∧ dom ◡𝐹 ⊆ 𝐵) → (◡𝐹 ↾ 𝐵) = ◡𝐹)
36	31, 34, 35	sylancr 588	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 ↾ 𝐵) = ◡𝐹)
37	36	fneq1d 6593	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → ((◡𝐹 ↾ 𝐵) Fn 𝐵 ↔ ◡𝐹 Fn 𝐵))
38	30, 37	bitr3id 285	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶 ↔ ◡𝐹 Fn 𝐵))
39	4, 38	bitr4d 282	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶))
40	39	pm5.32i 574	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴–1-1-onto→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶))
41		f1of 6782	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
42	41	pm4.71ri 560	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴–1-1-onto→𝐵))
43	7	fmpt 7064	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
44	43	anbi1i 625	. 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 303	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃!weu 2569  ∀wral 3052  ∃!wreu 3350   ⊆ wss 3903  ⟨cop 4588   class class class wbr 5100  {copab 5162   ↦ cmpt 5181  ^◡ccnv 5631  dom cdm 5632  ran crn 5633   ↾ cres 5634  Rel wrel 5637   Fn wfn 6495  ⟶wf 6496  –1-1-onto→wf1o 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507

This theorem is referenced by:  oaf1o  8500  xpf1o  9079  icoshftf1o  13402  fprodser  15884  dfod2  19505  gsummptf1o  19904  nbusgrf1o0  29454  cusgrfilem2  29542  numclwlk2lem2f1o  30466  f1mptrn  32724  ccatws1f1o  33043  gsummptf1od  33148  gsummptfsf1o  33153  xrmulc1cn  34107  poimirlem4  37872  poimirlem16  37884  poimirlem17  37885  poimirlem19  37887  poimirlem20  37888  isuspgrim0lem  48250  isuspgrim0  48251  isuspgrimlem  48252