Theorem funcnvmpt 6944

Description: Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.) (Proof shortened by Peter Mazsa, 24-Feb-2026.)

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 6064	. . . 4 ⊢ Rel ◡𝐹
2		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑦◡𝐹
3		funcnvmpt.2	. . . . . 6 ⊢ Ⅎ𝑥𝐹
4	3	nfcnv 5828	. . . . 5 ⊢ Ⅎ𝑥◡𝐹
5	2, 4	dffun6f 6508	. . . 4 ⊢ (Fun ◡𝐹 ↔ (Rel ◡𝐹 ∧ ∀𝑦∃*𝑥 𝑦◡𝐹𝑥))
6	1, 5	mpbiran 710	. . 3 ⊢ (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 𝑦◡𝐹𝑥)
7		vex 3445	. . . . . 6 ⊢ 𝑦 ∈ V
8		vex 3445	. . . . . 6 ⊢ 𝑥 ∈ V
9	7, 8	brcnv 5832	. . . . 5 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
10	9	mobii 2549	. . . 4 ⊢ (∃*𝑥 𝑦◡𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
11	10	albii 1821	. . 3 ⊢ (∀𝑦∃*𝑥 𝑦◡𝐹𝑥 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
12	6, 11	bitri 275	. 2 ⊢ (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
13		funcnvmpt.0	. . . . 5 ⊢ Ⅎ𝑥𝜑
14		funcnvmpt.3	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
15	14	funmpt2 6532	. . . . . . . . 9 ⊢ Fun 𝐹
16		funbrfv2b 6892	. . . . . . . . 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	14	dmmpt 6199	. . . . . . . . . . 11 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
19		funcnvmpt.4	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
20	19	elexd 3465	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
21	13, 20	ralrimia 3236	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
22		funcnvmpt.1	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐴
23	22	rabid2f 3431	. . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
24	21, 23	sylibr 234	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V})
25	18, 24	eqtr4id 2791	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝐴)
26	25	eleq2d 2823	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
27	26	anbi1d 632	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
28	17, 27	bitrid 283	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
29	28	bian1d 580	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦)))
30		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
31	14	fveq1i 6836	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)
32	22	fvmpt2f 6943	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
33	31, 32	eqtrid 2784	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝑥) = 𝐵)
34	30, 19, 33	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
35	34	eqeq2d 2748	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑦 = 𝐵))
36		eqcom 2744	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
37	26	biimpar 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
38		funbrfvb 6888	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
39	15, 37, 38	sylancr 588	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
40	36, 39	bitr3id 285	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
41	35, 40	bitr3d 281	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐵 ↔ 𝑥𝐹𝑦))
42	41	pm5.32da 579	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
43	29, 42, 28	3bitr4rd 312	. . . . 5 ⊢ (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
44	13, 43	mobid 2551	. . . 4 ⊢ (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
45		df-rmo 3351	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
46	44, 45	bitr4di 289	. . 3 ⊢ (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
47	46	albidv 1922	. 2 ⊢ (𝜑 → (∀𝑦∃*𝑥 𝑥𝐹𝑦 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
48	12, 47	bitrid 283	1 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  ∃*wmo 2538  Ⅎwnfc 2884  ∀wral 3052  ∃*wrmo 3350  {crab 3400  Vcvv 3441   class class class wbr 5099   ↦ cmpt 5180  ^◡ccnv 5624  dom cdm 5625  Rel wrel 5630  Fun wfun 6487  ‘cfv 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501

This theorem is referenced by:  funcnv5mpt  32727  disjqmap2  38998