Theorem funcnvmpt 28851

Description: Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.)

Hypotheses

Ref	Expression
funcnvmpt.0	⊢ Ⅎ𝑥𝜑
funcnvmpt.1	⊢ Ⅎ𝑥𝐴
funcnvmpt.2	⊢ Ⅎ𝑥𝐹
funcnvmpt.3	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
funcnvmpt.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
funcnvmpt	⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐹   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐹(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem funcnvmpt

Step	Hyp	Ref	Expression
1		relcnv 5422	. . . 4 ⊢ Rel ◡𝐹
2		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑦◡𝐹
3		funcnvmpt.2	. . . . . 6 ⊢ Ⅎ𝑥𝐹
4	3	nfcnv 5223	. . . . 5 ⊢ Ⅎ𝑥◡𝐹
5	2, 4	dffun6f 5818	. . . 4 ⊢ (Fun ◡𝐹 ↔ (Rel ◡𝐹 ∧ ∀𝑦∃*𝑥 𝑦◡𝐹𝑥))
6	1, 5	mpbiran 955	. . 3 ⊢ (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 𝑦◡𝐹𝑥)
7		vex 3176	. . . . . 6 ⊢ 𝑦 ∈ V
8		vex 3176	. . . . . 6 ⊢ 𝑥 ∈ V
9	7, 8	brcnv 5227	. . . . 5 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
10	9	mobii 2481	. . . 4 ⊢ (∃*𝑥 𝑦◡𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
11	10	albii 1737	. . 3 ⊢ (∀𝑦∃*𝑥 𝑦◡𝐹𝑥 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
12	6, 11	bitri 263	. 2 ⊢ (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
13		funcnvmpt.0	. . . . 5 ⊢ Ⅎ𝑥𝜑
14		funcnvmpt.3	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
15	14	funmpt2 5841	. . . . . . . . 9 ⊢ Fun 𝐹
16		funbrfv2b 6150	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑦)))
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ (𝑥𝐹𝑦 ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑦))
18		funcnvmpt.4	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
19		elex 3185	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
20	18, 19	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
21	20	ex 449	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ V))
22	13, 21	ralrimi 2940	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
23		funcnvmpt.1	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐴
24	23	rabid2f 28724	. . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
25	22, 24	sylibr 223	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V})
26	14	dmmpt 5547	. . . . . . . . . . 11 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
27	25, 26	syl6reqr 2663	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝐴)
28	27	eleq2d 2673	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
29	28	anbi1d 737	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
30	17, 29	syl5bb 271	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
31	30	bian1d 28690	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
32		simpr 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
33	14	fveq1i 6104	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)
34	23	fvmpt2f 6192	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
35	33, 34	syl5eq 2656	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝑥) = 𝐵)
36	32, 18, 35	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
37	36	eqeq2d 2620	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑦 = 𝐵))
38		eqcom 2617	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
39	28	biimpar 501	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
40		funbrfvb 6148	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
41	15, 39, 40	sylancr 694	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
42	38, 41	syl5bbr 273	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
43	37, 42	bitr3d 269	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐵 ↔ 𝑥𝐹𝑦))
44	43	pm5.32da 671	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
45	31, 44, 30	3bitr4rd 300	. . . . 5 ⊢ (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
46	13, 45	mobid 2477	. . . 4 ⊢ (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
47		df-rmo 2904	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
48	46, 47	syl6bbr 277	. . 3 ⊢ (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
49	48	albidv 1836	. 2 ⊢ (𝜑 → (∀𝑦∃*𝑥 𝑥𝐹𝑦 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
50	12, 49	syl5bb 271	1 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  ∃*wmo 2459  Ⅎwnfc 2738  ∀wral 2896  ∃*wrmo 2899  {crab 2900  Vcvv 3173   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038  Rel wrel 5043  Fun wfun 5798  ‘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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  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-ral 2901  df-rex 2902  df-rmo 2904  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-fv 5812

This theorem is referenced by:  funcnv5mpt  28852