Theorem funcnvmptOLD 29467

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 5503	. . . 4 ⊢ Rel ◡𝐹
2		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑦◡𝐹
3		funcnvmpt.2	. . . . . 6 ⊢ Ⅎ𝑥𝐹
4	3	nfcnv 5301	. . . . 5 ⊢ Ⅎ𝑥◡𝐹
5	2, 4	dffun6f 5902	. . . 4 ⊢ (Fun ◡𝐹 ↔ (Rel ◡𝐹 ∧ ∀𝑦∃*𝑥 𝑦◡𝐹𝑥))
6	1, 5	mpbiran 953	. . 3 ⊢ (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 𝑦◡𝐹𝑥)
7		vex 3203	. . . . . 6 ⊢ 𝑦 ∈ V
8		vex 3203	. . . . . 6 ⊢ 𝑥 ∈ V
9	7, 8	brcnv 5305	. . . . 5 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
10	9	mobii 2493	. . . 4 ⊢ (∃*𝑥 𝑦◡𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
11	10	albii 1747	. . 3 ⊢ (∀𝑦∃*𝑥 𝑦◡𝐹𝑥 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
12	6, 11	bitri 264	. 2 ⊢ (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
13		nfv 1843	. . 3 ⊢ Ⅎ𝑦𝜑
14		funcnvmpt.0	. . . 4 ⊢ Ⅎ𝑥𝜑
15		funmpt 5926	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
16		funcnvmpt.3	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
17	16	funeqi 5909	. . . . . . . . 9 ⊢ (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵))
18	15, 17	mpbir 221	. . . . . . . 8 ⊢ Fun 𝐹
19		funbrfv2b 6240	. . . . . . . 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 3212	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
23	21, 22	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
24	23	ex 450	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ V))
25	14, 24	ralrimi 2957	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
26		funcnvmpt.1	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐴
27	26	rabid2f 3119	. . . . . . . . . . 11 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
28	25, 27	sylibr 224	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V})
29	16	dmmpt 5630	. . . . . . . . . 10 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
30	28, 29	syl6reqr 2675	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
31	30	eleq2d 2687	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
32	31	anbi1d 741	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
33	20, 32	syl5bb 272	. . . . . 6 ⊢ (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
34	33	bian1d 29306	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
35	16	fveq1i 6192	. . . . . . . . . 10 ⊢ (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)
36		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
37	26	fvmpt2f 6283	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
38	36, 21, 37	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
39	35, 38	syl5eq 2668	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
40	39	eqeq2d 2632	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑦 = 𝐵))
41	31	biimpar 502	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
42		funbrfvb 6238	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
43	18, 41, 42	sylancr 695	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
44		eqcom 2629	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
45	44	bibi1i 328	. . . . . . . . . 10 ⊢ (((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦) ↔ (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
46	45	imbi2i 326	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦)))
47	43, 46	mpbi 220	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
48	40, 47	bitr3d 270	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐵 ↔ 𝑥𝐹𝑦))
49	48	ex 450	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐵 ↔ 𝑥𝐹𝑦)))
50	49	pm5.32d 671	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
51	34, 50, 33	3bitr4rd 301	. . . 4 ⊢ (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
52	14, 51	mobid 2489	. . 3 ⊢ (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
53	13, 52	albid 2090	. 2 ⊢ (𝜑 → (∀𝑦∃*𝑥 𝑥𝐹𝑦 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
54	12, 53	syl5bb 272	1 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  ∃*wmo 2471  Ⅎwnfc 2751  ∀wral 2912  {crab 2916  Vcvv 3200   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113  dom cdm 5114  Rel wrel 5119  Fun wfun 5882  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by: (None)