Theorem dffun4f 5204

Description: Definition of function like dffun4 5199 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.)

Hypotheses

Ref	Expression
dffun4f.1	⊢ Ⅎ𝑥𝐴
dffun4f.2	⊢ Ⅎ𝑦𝐴
dffun4f.3	⊢ Ⅎ𝑧𝐴

Assertion

Ref	Expression
dffun4f	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧)))

Distinct variable group:   𝑥,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem dffun4f

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun4f.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		dffun4f.2	. . 3 ⊢ Ⅎ𝑦𝐴
3	1, 2	dffun6f 5201	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
4		nfcv 2308	. . . . . . 7 ⊢ Ⅎ𝑦𝑥
5		nfcv 2308	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
6	4, 2, 5	nfbr 4028	. . . . . 6 ⊢ Ⅎ𝑦 𝑥𝐴𝑤
7		breq2 3986	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑥𝐴𝑦 ↔ 𝑥𝐴𝑤))
8	6, 7	mo4f 2074	. . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑦∀𝑤((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑤) → 𝑦 = 𝑤))
9		nfv 1516	. . . . . . 7 ⊢ Ⅎ𝑤((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)
10		nfcv 2308	. . . . . . . . . 10 ⊢ Ⅎ𝑧𝑥
11		dffun4f.3	. . . . . . . . . 10 ⊢ Ⅎ𝑧𝐴
12		nfcv 2308	. . . . . . . . . 10 ⊢ Ⅎ𝑧𝑦
13	10, 11, 12	nfbr 4028	. . . . . . . . 9 ⊢ Ⅎ𝑧 𝑥𝐴𝑦
14		nfcv 2308	. . . . . . . . . 10 ⊢ Ⅎ𝑧𝑤
15	10, 11, 14	nfbr 4028	. . . . . . . . 9 ⊢ Ⅎ𝑧 𝑥𝐴𝑤
16	13, 15	nfan 1553	. . . . . . . 8 ⊢ Ⅎ𝑧(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑤)
17		nfv 1516	. . . . . . . 8 ⊢ Ⅎ𝑧 𝑦 = 𝑤
18	16, 17	nfim 1560	. . . . . . 7 ⊢ Ⅎ𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑤) → 𝑦 = 𝑤)
19		breq2 3986	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑥𝐴𝑧 ↔ 𝑥𝐴𝑤))
20	19	anbi2d 460	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑥𝐴𝑤)))
21		equequ2 1701	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑦 = 𝑧 ↔ 𝑦 = 𝑤))
22	20, 21	imbi12d 233	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑤) → 𝑦 = 𝑤)))
23	9, 18, 22	cbval 1742	. . . . . 6 ⊢ (∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑤) → 𝑦 = 𝑤))
24	23	albii 1458	. . . . 5 ⊢ (∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦∀𝑤((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑤) → 𝑦 = 𝑤))
25	8, 24	bitr4i 186	. . . 4 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
26	25	albii 1458	. . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
27	26	anbi2i 453	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
28		df-br 3983	. . . . . . 7 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
29		df-br 3983	. . . . . . 7 ⊢ (𝑥𝐴𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
30	28, 29	anbi12i 456	. . . . . 6 ⊢ ((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
31	30	imbi1i 237	. . . . 5 ⊢ (((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
32	31	2albii 1459	. . . 4 ⊢ (∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
33	32	albii 1458	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
34	33	anbi2i 453	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧)))
35	3, 27, 34	3bitri 205	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341  ∃*wmo 2015   ∈ wcel 2136  Ⅎwnfc 2295  ⟨cop 3579   class class class wbr 3982  Rel wrel 4609  Fun wfun 5182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-cnv 4612  df-co 4613  df-fun 5190

This theorem is referenced by: (None)