Theorem funcnv4mpt 32512

Description: Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-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
funcnv4mpt	⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)))

Distinct variable groups:   𝑖,𝑗,𝑥   𝐴,𝑖,𝑗   𝐵,𝑖,𝑗   𝑖,𝐹   𝑥,𝑉   𝜑,𝑖,𝑗

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

Proof of Theorem funcnv4mpt

Step	Hyp	Ref	Expression
1		nfv 1909	. 2 ⊢ Ⅎ𝑖𝜑
2		nfcv 2892	. 2 ⊢ Ⅎ𝑖𝐴
3		nfcv 2892	. 2 ⊢ Ⅎ𝑖𝐹
4		funcnvmpt.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5		funcnvmpt.1	. . . 4 ⊢ Ⅎ𝑥𝐴
6		nfcv 2892	. . . 4 ⊢ Ⅎ𝑖𝐵
7		nfcsb1v 3915	. . . 4 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵
8		csbeq1a 3904	. . . 4 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵)
9	5, 2, 6, 7, 8	cbvmptf 5257	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑖 ∈ 𝐴 ↦ ⦋𝑖 / 𝑥⦌𝐵)
10	4, 9	eqtri 2753	. 2 ⊢ 𝐹 = (𝑖 ∈ 𝐴 ↦ ⦋𝑖 / 𝑥⦌𝐵)
11		funcnvmpt.4	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
12	11	sbimi 2069	. . 3 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑖 / 𝑥]𝐵 ∈ 𝑉)
13		funcnvmpt.0	. . . . 5 ⊢ Ⅎ𝑥𝜑
14		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑥𝑖
15	14, 5	nfel 2907	. . . . 5 ⊢ Ⅎ𝑥 𝑖 ∈ 𝐴
16	13, 15	nfan 1894	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ 𝐴)
17		eleq1w 2808	. . . . 5 ⊢ (𝑥 = 𝑖 → (𝑥 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴))
18	17	anbi2d 628	. . . 4 ⊢ (𝑥 = 𝑖 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴)))
19	16, 18	sbiev 2303	. . 3 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴))
20		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑥𝑉
21	7, 20	nfel 2907	. . . 4 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉
22	8	eleq1d 2810	. . . 4 ⊢ (𝑥 = 𝑖 → (𝐵 ∈ 𝑉 ↔ ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉))
23	21, 22	sbiev 2303	. . 3 ⊢ ([𝑖 / 𝑥]𝐵 ∈ 𝑉 ↔ ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉)
24	12, 19, 23	3imtr3i 290	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉)
25		csbeq1 3893	. 2 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵)
26	1, 2, 3, 10, 24, 25	funcnv5mpt 32511	1 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533  Ⅎwnf 1777  [wsb 2059   ∈ wcel 2098  Ⅎwnfc 2875   ≠ wne 2930  ∀wral 3051  ⦋csb 3890   ↦ cmpt 5231  ^◡ccnv 5676  Fun wfun 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6499  df-fun 6549  df-fn 6550  df-fv 6555

This theorem is referenced by:  disjdsct  32539