Theorem f1ompt 7086

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 6685	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		dff1o4 6809	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
3	2	baib 543	. . . . 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 6642	. . . . . 6 ⊢ ((◡𝐹 ↾ 𝐵) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑧 𝑦◡𝐹𝑧)
6		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
7		fmpt.1	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
8		nfmpt1 5196	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
9	7, 8	nfcxfr 2921	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
10		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑦
11	6, 9, 10	nfbr 5144	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧𝐹𝑦
12		nfv 1933	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)
13		breq1 5100	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧𝐹𝑦 ↔ 𝑥𝐹𝑦))
14		df-mpt 5179	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
15	7, 14	eqtri 2784	. . . . . . . . . . . 12 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
16	15	breqi 5103	. . . . . . . . . . 11 ⊢ (𝑥𝐹𝑦 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}𝑦)
17		df-br 5098	. . . . . . . . . . . 12 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)})
18		opabidw 5491	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
19	17, 18	bitri 277	. . . . . . . . . . 11 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
20	16, 19	bitri 277	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
21	13, 20	bitrdi 289	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)))
22	11, 12, 21	cbveuw 2632	. . . . . . . 8 ⊢ (∃!𝑧 𝑧𝐹𝑦 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
23		vex 3457	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
24		vex 3457	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
25	23, 24	brcnv 5850	. . . . . . . . 9 ⊢ (𝑦◡𝐹𝑧 ↔ 𝑧𝐹𝑦)
26	25	eubii 2611	. . . . . . . 8 ⊢ (∃!𝑧 𝑦◡𝐹𝑧 ↔ ∃!𝑧 𝑧𝐹𝑦)
27		df-reu 3367	. . . . . . . 8 ⊢ (∃!𝑥 ∈ 𝐴 𝑦 = 𝐶 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
28	22, 26, 27	3bitr4i 305	. . . . . . 7 ⊢ (∃!𝑧 𝑦◡𝐹𝑧 ↔ ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶)
29	28	ralbii 3107	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ∃!𝑧 𝑦◡𝐹𝑧 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶)
30	5, 29	bitri 277	. . . . 5 ⊢ ((◡𝐹 ↾ 𝐵) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶)
31		relcnv 6088	. . . . . . 7 ⊢ Rel ◡𝐹
32		df-rn 5654	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
33		frn 6693	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
34	32, 33	eqsstrrid 3973	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → dom ◡𝐹 ⊆ 𝐵)
35		relssres 6004	. . . . . . 7 ⊢ ((Rel ◡𝐹 ∧ dom ◡𝐹 ⊆ 𝐵) → (◡𝐹 ↾ 𝐵) = ◡𝐹)
36	31, 34, 35	sylancr 596	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 ↾ 𝐵) = ◡𝐹)
37	36	fneq1d 6608	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → ((◡𝐹 ↾ 𝐵) Fn 𝐵 ↔ ◡𝐹 Fn 𝐵))
38	30, 37	bitr3id 287	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶 ↔ ◡𝐹 Fn 𝐵))
39	4, 38	bitr4d 284	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶))
40	39	pm5.32i 582	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴–1-1-onto→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶))
41		f1of 6800	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
42	41	pm4.71ri 568	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴–1-1-onto→𝐵))
43	7	fmpt 7085	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
44	43	anbi1i 633	. 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 305	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃!weu 2594  ∀wral 3075  ∃!wreu 3364   ⊆ wss 3902  ⟨cop 4585   class class class wbr 5097  {copab 5159   ↦ cmpt 5178  ^◡ccnv 5642  dom cdm 5643  ran crn 5644   ↾ cres 5645  Rel wrel 5648   Fn wfn 6510  ⟶wf 6511  –1-1-onto→wf1o 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522

This theorem is referenced by:  oaf1o  8525  xpf1o  9104  icoshftf1o  13471  fprodser  15969  dfod2  19594  gsummptf1o  19993  nbusgrf1o0  29526  cusgrfilem2  29613  numclwlk2lem2f1o  30537  f1mptrn  32797  ccatws1f1o  33089  gsummptf1od  33195  gsummptfsf1o  33200  xrmulc1cn  34187  poimirlem4  38083  poimirlem16  38095  poimirlem17  38096  poimirlem19  38098  poimirlem20  38099  isuspgrim0lem  48475  isuspgrim0  48476  isuspgrimlem  48477