Theorem dffun2 6547

Description: Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2178, ax-12 2215. (Revised by SN, 19-Dec-2024.) Avoid ax-11 2194. (Revised by BTernaryTau, 29-Dec-2024.)

Assertion

Ref	Expression
dffun2	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))

Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem dffun2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fun 6539	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))
2		cotrg 6109	. . . 4 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧))
3		breq1 5110	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦◡𝐴𝑥 ↔ 𝑤◡𝐴𝑥))
4	3	anbi1d 643	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ (𝑤◡𝐴𝑥 ∧ 𝑥𝐴𝑧)))
5		breq1 5110	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑦 I 𝑧 ↔ 𝑤 I 𝑧))
6	4, 5	imbi12d 347	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑤◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑤 I 𝑧)))
7	6	albidv 1953	. . . . . 6 ⊢ (𝑦 = 𝑤 → (∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑧((𝑤◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑤 I 𝑧)))
8		breq2 5111	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑦◡𝐴𝑥 ↔ 𝑦◡𝐴𝑤))
9		breq1 5110	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥𝐴𝑧 ↔ 𝑤𝐴𝑧))
10	8, 9	anbi12d 644	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ (𝑦◡𝐴𝑤 ∧ 𝑤𝐴𝑧)))
11	10	imbi1d 344	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑦◡𝐴𝑤 ∧ 𝑤𝐴𝑧) → 𝑦 I 𝑧)))
12	11	albidv 1953	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑧((𝑦◡𝐴𝑤 ∧ 𝑤𝐴𝑧) → 𝑦 I 𝑧)))
13	7, 12	alcomw 2078	. . . . 5 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧))
14		vex 3457	. . . . . . . . 9 ⊢ 𝑦 ∈ V
15		vex 3457	. . . . . . . . 9 ⊢ 𝑥 ∈ V
16	14, 15	brcnv 5866	. . . . . . . 8 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
17	16	anbi1i 636	. . . . . . 7 ⊢ ((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧))
18		vex 3457	. . . . . . . 8 ⊢ 𝑧 ∈ V
19	18	ideq 5836	. . . . . . 7 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧)
20	17, 19	imbi12i 353	. . . . . 6 ⊢ (((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
21	20	3albii 1854	. . . . 5 ⊢ (∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
22	13, 21	bitri 278	. . . 4 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
23	2, 22	bitri 278	. . 3 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
24	23	anbi2i 635	. 2 ⊢ ((Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
25	1, 24	bitri 278	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   ⊆ wss 3902   class class class wbr 5107   I cid 5553  ^◡ccnv 5658   ∘ ccom 5663  Rel wrel 5664  Fun wfun 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-fun 6539

This theorem is used by:  dffun6  6548  dffun4  6550  fundif  6586  fliftfun  7317  frrlem9  8297  fprlem1  8303  frrlem15  9743  fpwwe2lem10  10653  fclim  15644  invfun  17859  lmfun  23612  ulmdm  26636  fundmpss  36354  fununiq  36356  fnsingle  36504  funimage  36513  funpartfun  36530  functhincfun  50383