Theorem funcnv5mpt 29469

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 29468	. 2 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
7		nne 2798	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
8		eqvincg 3329	. . . . . . . . . 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	syl5bb 272	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝐵 ≠ 𝐶 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶)))
11	10	imbi1d 331	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((¬ 𝐵 ≠ 𝐶 → 𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
12		orcom 402	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ (𝐵 ≠ 𝐶 ∨ 𝑥 = 𝑧))
13		df-or 385	. . . . . . . 8 ⊢ ((𝐵 ≠ 𝐶 ∨ 𝑥 = 𝑧) ↔ (¬ 𝐵 ≠ 𝐶 → 𝑥 = 𝑧))
14	12, 13	bitri 264	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ (¬ 𝐵 ≠ 𝐶 → 𝑥 = 𝑧))
15		19.23v 1902	. . . . . . 7 ⊢ (∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
16	11, 14, 15	3bitr4g 303	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
17	16	ralbidv 2986	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑧 ∈ 𝐴 ∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
18		ralcom4 3224	. . . . 5 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧) ↔ ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
19	17, 18	syl6bb 276	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
20	1, 19	ralbida 2982	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
21		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑧𝐴
22		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐶
23		funcnv5mpt.1	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐶)
24	23	eqeq2d 2632	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶))
25	2, 21, 22, 24	rmo4f 29337	. . . . 5 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
26	25	albii 1747	. . . 4 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∀𝑦∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
27		ralcom4 3224	. . . 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 267	. . 3 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
29	20, 28	syl6bbr 278	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
30	6, 29	bitr4d 271	1 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704  Ⅎwnf 1708   ∈ wcel 1990  Ⅎwnfc 2751   ≠ wne 2794  ∀wral 2912  ∃*wrmo 2915   ↦ cmpt 4729  ^◡ccnv 5113  Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  funcnv4mpt  29470