Theorem funcnv5mpt 31881

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 31880	. 2 ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
7		nne 2945	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
8		eqvincg 3636	. . . . . . . . . 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 342	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((¬ 𝐵 ≠ 𝐶 → 𝑥 = 𝑧) ↔ (∃𝑦(𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
12		orcom 869	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ (𝐵 ≠ 𝐶 ∨ 𝑥 = 𝑧))
13		df-or 847	. . . . . . . 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 1946	. . . . . . 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 3178	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑧 ∈ 𝐴 ∀𝑦((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
18		ralcom4 3284	. . . . 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 3268	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧)))
21		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑧𝐴
22		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐶
23		funcnv5mpt.1	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐶)
24	23	eqeq2d 2744	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶))
25	2, 21, 22, 24	rmo4f 3731	. . . . 5 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
26	25	albii 1822	. . . 4 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∀𝑦∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑦 = 𝐵 ∧ 𝑦 = 𝐶) → 𝑥 = 𝑧))
27		ralcom4 3284	. . . 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 397   ∨ wo 846  ∀wal 1540   = wceq 1542  ∃wex 1782  Ⅎwnf 1786   ∈ wcel 2107  Ⅎwnfc 2884   ≠ wne 2941  ∀wral 3062  ∃*wrmo 3376   ↦ cmpt 5231  ^◡ccnv 5675  Fun wfun 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-fv 6549

This theorem is referenced by:  funcnv4mpt  31882