Theorem funcnv4mpt 32979

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
funcnv5mpt.0	⊢ Ⅎ𝑥𝜑
funcnv5mpt.1	⊢ Ⅎ𝑥𝐴
funcnv5mpt.2	⊢ Ⅎ𝑥𝐹
funcnv5mpt.3	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
funcnv5mpt.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 1942	. 2 ⊢ Ⅎ𝑖𝜑
2		nfcv 2923	. 2 ⊢ Ⅎ𝑖𝐴
3		nfcv 2923	. 2 ⊢ Ⅎ𝑖𝐹
4		funcnv5mpt.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5		funcnv5mpt.1	. . . 4 ⊢ Ⅎ𝑥𝐴
6		nfcv 2923	. . . 4 ⊢ Ⅎ𝑖𝐵
7		nfcsb1v 3876	. . . 4 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵
8		csbeq1a 3866	. . . 4 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵)
9	5, 2, 6, 7, 8	cbvmptf 5210	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑖 ∈ 𝐴 ↦ ⦋𝑖 / 𝑥⦌𝐵)
10	4, 9	eqtri 2784	. 2 ⊢ 𝐹 = (𝑖 ∈ 𝐴 ↦ ⦋𝑖 / 𝑥⦌𝐵)
11		funcnv5mpt.4	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
12	11	sbimi 2106	. . 3 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑖 / 𝑥]𝐵 ∈ 𝑉)
13		funcnv5mpt.0	. . . . 5 ⊢ Ⅎ𝑥𝜑
14		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥𝑖
15	14, 5	nfel 2937	. . . . 5 ⊢ Ⅎ𝑥 𝑖 ∈ 𝐴
16	13, 15	nfan 1927	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ 𝐴)
17		eleq1w 2844	. . . . 5 ⊢ (𝑥 = 𝑖 → (𝑥 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴))
18	17	anbi2d 641	. . . 4 ⊢ (𝑥 = 𝑖 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴)))
19	16, 18	sbiev 2345	. . 3 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴))
20		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑥𝑉
21	7, 20	nfel 2937	. . . 4 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉
22	8	eleq1d 2846	. . . 4 ⊢ (𝑥 = 𝑖 → (𝐵 ∈ 𝑉 ↔ ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉))
23	21, 22	sbiev 2345	. . 3 ⊢ ([𝑖 / 𝑥]𝐵 ∈ 𝑉 ↔ ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉)
24	12, 19, 23	3imtr3i 294	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉)
25		csbeq1 3855	. 2 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵)
26	1, 2, 3, 10, 24, 25	funcnv5mpt 32978	1 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1568  Ⅎwnf 1811  [wsb 2094   ∈ wcel 2141  Ⅎwnfc 2908   ≠ wne 2956  ∀wral 3077  ⦋csb 3852   ↦ cmpt 5191  ^◡ccnv 5660  Fun wfun 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is referenced by:  disjdsct  33014