Theorem funcnv5mpt 32402

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
funcnvmpt.0	⊢ Ⅎ𝑥𝜑
funcnvmpt.1	⊢ Ⅎ𝑥𝐴
funcnvmpt.2	⊢ Ⅎ𝑥𝐹
funcnvmpt.3	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
funcnvmpt.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
funcnv5mpt.1	⊢ (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		funcnvmpt.0	. . 3 ⊢ Ⅎ𝑥𝜑
2		funcnvmpt.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		funcnvmpt.2	. . 3 ⊢ Ⅎ𝑥𝐹
4		funcnvmpt.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5		funcnvmpt.4	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
6	1, 2, 3, 4, 5	funcnvmpt 32401	. 2 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
7		nne 2938	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
8		eqvincg 3631	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑉 → (𝐵 = 𝐶 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶)))
9	5, 8	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = 𝐶 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶)))
10	7, 9	bitrid 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝐵 ≠ 𝐶 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶)))
11	10	imbi1d 341	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((¬ 𝐵 ≠ 𝐶 → 𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
12		orcom 867	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ (𝐵 ≠ 𝐶 ∨ 𝑥 = 𝑧))
13		df-or 845	. . . . . . . 8 ⊢ ((𝐵 ≠ 𝐶 ∨ 𝑥 = 𝑧) ↔ (¬ 𝐵 ≠ 𝐶 → 𝑥 = 𝑧))
14	12, 13	bitri 275	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ (¬ 𝐵 ≠ 𝐶 → 𝑥 = 𝑧))
15		19.23v 1937	. . . . . . 7 ⊢ (∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
16	11, 14, 15	3bitr4g 314	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
17	16	ralbidv 3171	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑧 ∈ 𝐴 ∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
18		ralcom4 3277	. . . . 5 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
19	17, 18	bitrdi 287	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
20	1, 19	ralbida 3261	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
21		nfcv 2897	. . . . . 6 ⊢ Ⅎ𝑧𝐴
22		nfv 1909	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐶
23		funcnv5mpt.1	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐶)
24	23	eqeq2d 2737	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶))
25	2, 21, 22, 24	rmo4f 3726	. . . . 5 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
26	25	albii 1813	. . . 4 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∀𝑦∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
27		ralcom4 3277	. . . 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 278	. . 3 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
29	20, 28	bitr4di 289	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
30	6, 29	bitr4d 282	1 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844  ∀wal 1531   = wceq 1533  ∃wex 1773  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2877   ≠ wne 2934  ∀wral 3055  ∃*wrmo 3369   ↦ cmpt 5224  ^◡ccnv 5668  Fun wfun 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-fv 6545

This theorem is referenced by:  funcnv4mpt  32403