Theorem funcnv5mpt 33041

Description: Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-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)
funcnv5mpt.5	⊢ (𝑥 = 𝑧 → 𝐵 = 𝐶)

Assertion

Ref	Expression
funcnv5mpt	⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶)))

Distinct variable groups:   𝑥,𝑧   𝜑,𝑧   𝑧,𝐴   𝑧,𝐵   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑧)   𝐹(𝑥, 𝑧)   𝑉(𝑥, 𝑧)

Proof of Theorem funcnv5mpt

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funcnv5mpt.0	. . 3 ⊢ Ⅎ𝑥𝜑
2		funcnv5mpt.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		funcnv5mpt.2	. . 3 ⊢ Ⅎ𝑥𝐹
4		funcnv5mpt.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5		funcnv5mpt.4	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
6	1, 2, 3, 4, 5	funcnvmpt 6995	. 2 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
7		nne 2964	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
8		eqvincg 3609	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑉 → (𝐵 = 𝐶 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶)))
9	5, 8	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = 𝐶 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶)))
10	7, 9	bitrid 286	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝐵 ≠ 𝐶 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶)))
11	10	imbi1d 344	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((¬ 𝐵 ≠ 𝐶 → 𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
12		orcom 884	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ (𝐵 ≠ 𝐶 ∨ 𝑥 = 𝑧))
13		df-or 862	. . . . . . . 8 ⊢ ((𝐵 ≠ 𝐶 ∨ 𝑥 = 𝑧) ↔ (¬ 𝐵 ≠ 𝐶 → 𝑥 = 𝑧))
14	12, 13	bitri 278	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ (¬ 𝐵 ≠ 𝐶 → 𝑥 = 𝑧))
15		19.23v 1975	. . . . . . 7 ⊢ (∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
16	11, 14, 15	3bitr4g 317	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
17	16	ralbidv 3190	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑧 ∈ 𝐴 ∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
18		ralcom4 3293	. . . . 5 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
19	17, 18	bitrdi 290	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
20	1, 19	ralbida 3278	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
21		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑧𝐴
22		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐶
23		funcnv5mpt.5	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐶)
24	23	eqeq2d 2776	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶))
25	2, 21, 22, 24	rmo4f 3700	. . . . 5 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
26	25	albii 1852	. . . 4 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∀𝑦∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
27		ralcom4 3293	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ ∀𝑦∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
28	26, 27	bitr4i 281	. . 3 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
29	20, 28	bitr4di 292	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
30	6, 29	bitr4d 285	1 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570  ∃wex 1812  Ⅎwnf 1816   ∈ wcel 2146  Ⅎwnfc 2912   ≠ wne 2960  ∀wral 3081  ∃*wrmo 3370   ↦ cmpt 5194  ^◡ccnv 5662  Fun wfun 6534

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548

This theorem is used by:  funcnv4mpt  33042