Theorem funcnv4mpt 32728

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 1916	. 2 ⊢ Ⅎ𝑖𝜑
2		nfcv 2899	. 2 ⊢ Ⅎ𝑖𝐴
3		nfcv 2899	. 2 ⊢ Ⅎ𝑖𝐹
4		funcnv5mpt.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5		funcnv5mpt.1	. . . 4 ⊢ Ⅎ𝑥𝐴
6		nfcv 2899	. . . 4 ⊢ Ⅎ𝑖𝐵
7		nfcsb1v 3874	. . . 4 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵
8		csbeq1a 3864	. . . 4 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵)
9	5, 2, 6, 7, 8	cbvmptf 5199	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑖 ∈ 𝐴 ↦ ⦋𝑖 / 𝑥⦌𝐵)
10	4, 9	eqtri 2760	. 2 ⊢ 𝐹 = (𝑖 ∈ 𝐴 ↦ ⦋𝑖 / 𝑥⦌𝐵)
11		funcnv5mpt.4	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
12	11	sbimi 2080	. . 3 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑖 / 𝑥]𝐵 ∈ 𝑉)
13		funcnv5mpt.0	. . . . 5 ⊢ Ⅎ𝑥𝜑
14		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥𝑖
15	14, 5	nfel 2914	. . . . 5 ⊢ Ⅎ𝑥 𝑖 ∈ 𝐴
16	13, 15	nfan 1901	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ 𝐴)
17		eleq1w 2820	. . . . 5 ⊢ (𝑥 = 𝑖 → (𝑥 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴))
18	17	anbi2d 631	. . . 4 ⊢ (𝑥 = 𝑖 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴)))
19	16, 18	sbiev 2320	. . 3 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴))
20		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑥𝑉
21	7, 20	nfel 2914	. . . 4 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉
22	8	eleq1d 2822	. . . 4 ⊢ (𝑥 = 𝑖 → (𝐵 ∈ 𝑉 ↔ ⦋𝑖 / 𝑥⦌𝐵 ∈ 𝑉))
23	21, 22	sbiev 2320	. . 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 3853	. 2 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵)
26	1, 2, 3, 10, 24, 25	funcnv5mpt 32727	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 848   = wceq 1542  Ⅎwnf 1785  [wsb 2068   ∈ wcel 2114  Ⅎwnfc 2884   ≠ wne 2933  ∀wral 3052  ⦋csb 3850   ↦ cmpt 5180  ^◡ccnv 5624  Fun wfun 6487

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:  disjdsct  32763