Theorem funcnvmptOLD 28850

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

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
funcnvmptOLD	⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥(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 funcnvmptOLD

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		nfv 1830	. . 3 ⊢ Ⅎ𝑦𝜑
14		funcnvmpt.0	. . . 4 ⊢ Ⅎ𝑥𝜑
15		funmpt 5840	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
16		funcnvmpt.3	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
17	16	funeqi 5824	. . . . . . . . 9 ⊢ (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵))
18	15, 17	mpbir 220	. . . . . . . 8 ⊢ Fun 𝐹
19		funbrfv2b 6150	. . . . . . . 8 ⊢ (Fun 𝐹 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑦)))
20	18, 19	ax-mp 5	. . . . . . 7 ⊢ (𝑥𝐹𝑦 ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑦))
21		funcnvmpt.4	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
22		elex 3185	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
23	21, 22	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
24	23	ex 449	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ V))
25	14, 24	ralrimi 2940	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
26		funcnvmpt.1	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐴
27	26	rabid2f 28724	. . . . . . . . . . 11 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
28	25, 27	sylibr 223	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V})
29	16	dmmpt 5547	. . . . . . . . . 10 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
30	28, 29	syl6reqr 2663	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
31	30	eleq2d 2673	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
32	31	anbi1d 737	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
33	20, 32	syl5bb 271	. . . . . 6 ⊢ (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
34	33	bian1d 28690	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
35	16	fveq1i 6104	. . . . . . . . . 10 ⊢ (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)
36		simpr 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
37	26	fvmpt2f 6192	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
38	36, 21, 37	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
39	35, 38	syl5eq 2656	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
40	39	eqeq2d 2620	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑦 = 𝐵))
41	31	biimpar 501	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
42		funbrfvb 6148	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
43	18, 41, 42	sylancr 694	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
44		eqcom 2617	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
45	44	bibi1i 327	. . . . . . . . . 10 ⊢ (((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦) ↔ (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
46	45	imbi2i 325	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦)))
47	43, 46	mpbi 219	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
48	40, 47	bitr3d 269	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐵 ↔ 𝑥𝐹𝑦))
49	48	ex 449	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐵 ↔ 𝑥𝐹𝑦)))
50	49	pm5.32d 669	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
51	34, 50, 33	3bitr4rd 300	. . . 4 ⊢ (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
52	14, 51	mobid 2477	. . 3 ⊢ (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
53	13, 52	albid 2077	. 2 ⊢ (𝜑 → (∀𝑦∃*𝑥 𝑥𝐹𝑦 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
54	12, 53	syl5bb 271	1 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥(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  {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-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: (None)