Theorem funcnv4mpt 32641

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 1915	. 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 3872	. . . 4 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵
8		csbeq1a 3862	. . . 4 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵)
9	5, 2, 6, 7, 8	cbvmptf 5189	. . 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 2076	. . 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 1900	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ 𝐴)
17		eleq1w 2812	. . . . 5 ⊢ (𝑥 = 𝑖 → (𝑥 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴))
18	17	anbi2d 630	. . . 4 ⊢ (𝑥 = 𝑖 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴)))
19	16, 18	sbiev 2314	. . 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 2814	. . . 4 ⊢ (𝑥 = 𝑖 → (𝐵 ∈ 𝑉 ↔ ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉))
23	21, 22	sbiev 2314	. . 3 ⊢ ([𝑖 / 𝑥]𝐵 ∈ 𝑉 ↔ ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉)
24	12, 19, 23	3imtr3i 291	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉)
25		csbeq1 3851	. 2 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵)
26	1, 2, 3, 10, 24, 25	funcnv5mpt 32640	1 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541  Ⅎwnf 1784  [wsb 2066   ∈ wcel 2110  Ⅎwnfc 2877   ≠ wne 2926  ∀wral 3045  ⦋csb 3848   ↦ cmpt 5170  ^◡ccnv 5613  Fun wfun 6471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-fv 6485

This theorem is referenced by:  disjdsct  32674