Theorem funcnvmpt 30406

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 5962	. . . 4 ⊢ Rel ◡𝐹
2		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑦◡𝐹
3		funcnvmpt.2	. . . . . 6 ⊢ Ⅎ𝑥𝐹
4	3	nfcnv 5744	. . . . 5 ⊢ Ⅎ𝑥◡𝐹
5	2, 4	dffun6f 6364	. . . 4 ⊢ (Fun ◡𝐹 ↔ (Rel ◡𝐹 ∧ ∀𝑦∃*𝑥 𝑦◡𝐹𝑥))
6	1, 5	mpbiran 707	. . 3 ⊢ (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 𝑦◡𝐹𝑥)
7		vex 3498	. . . . . 6 ⊢ 𝑦 ∈ V
8		vex 3498	. . . . . 6 ⊢ 𝑥 ∈ V
9	7, 8	brcnv 5748	. . . . 5 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
10	9	mobii 2627	. . . 4 ⊢ (∃*𝑥 𝑦◡𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
11	10	albii 1816	. . 3 ⊢ (∀𝑦∃*𝑥 𝑦◡𝐹𝑥 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
12	6, 11	bitri 277	. 2 ⊢ (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
13		funcnvmpt.0	. . . . 5 ⊢ Ⅎ𝑥𝜑
14		funcnvmpt.3	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
15	14	funmpt2 6389	. . . . . . . . 9 ⊢ Fun 𝐹
16		funbrfv2b 6718	. . . . . . . . 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	18	elexd 3515	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
20	19	ex 415	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ V))
21	13, 20	ralrimi 3216	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
22		funcnvmpt.1	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐴
23	22	rabid2f 3383	. . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
24	21, 23	sylibr 236	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V})
25	14	dmmpt 6089	. . . . . . . . . . 11 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
26	24, 25	syl6reqr 2875	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝐴)
27	26	eleq2d 2898	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
28	27	anbi1d 631	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
29	17, 28	syl5bb 285	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
30	29	bian1d 30218	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
31		simpr 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
32	14	fveq1i 6666	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)
33	22	fvmpt2f 6764	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
34	32, 33	syl5eq 2868	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝑥) = 𝐵)
35	31, 18, 34	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
36	35	eqeq2d 2832	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑦 = 𝐵))
37		eqcom 2828	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
38	27	biimpar 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
39		funbrfvb 6715	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
40	15, 38, 39	sylancr 589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
41	37, 40	syl5bbr 287	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
42	36, 41	bitr3d 283	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐵 ↔ 𝑥𝐹𝑦))
43	42	pm5.32da 581	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
44	30, 43, 29	3bitr4rd 314	. . . . 5 ⊢ (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
45	13, 44	mobid 2630	. . . 4 ⊢ (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
46		df-rmo 3146	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
47	45, 46	syl6bbr 291	. . 3 ⊢ (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
48	47	albidv 1917	. 2 ⊢ (𝜑 → (∀𝑦∃*𝑥 𝑥𝐹𝑦 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
49	12, 48	syl5bb 285	1 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533  Ⅎwnf 1780   ∈ wcel 2110  ∃*wmo 2616  Ⅎwnfc 2961  ∀wral 3138  ∃*wrmo 3141  {crab 3142  Vcvv 3495   class class class wbr 5059   ↦ cmpt 5139  ^◡ccnv 5549  dom cdm 5550  Rel wrel 5555  Fun wfun 6344  ‘cfv 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-fv 6358

This theorem is referenced by:  funcnv5mpt  30407