Theorem funcnv5mpt 33309

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 6996	. 2 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
7		nne 2959	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
8		eqvincg 3601	. . . . . . . . . 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 1974	. . . . . . 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 3185	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑧 ∈ 𝐴 ∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
18		ralcom4 3288	. . . . 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 3273	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
21		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑧𝐴
22		nfv 1946	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐶
23		funcnv5mpt.5	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐶)
24	23	eqeq2d 2771	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶))
25	2, 21, 22, 24	rmo4f 3692	. . . . 5 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
26	25	albii 1851	. . . 4 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∀𝑦∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
27		ralcom4 3288	. . . 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 1567   = wceq 1569  ∃wex 1811  Ⅎwnf 1815   ∈ wcel 2144  Ⅎwnfc 2907   ≠ wne 2955  ∀wral 3076  ∃*wrmo 3364   ↦ cmpt 5185  ^◡ccnv 5649  Fun wfun 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-fv 6547

This theorem is used by:  funcnv4mpt  33310