Theorem dffun2 6553

Description: Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2136, ax-12 2170. (Revised by SN, 19-Dec-2024.) Avoid ax-11 2153. (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 6545	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))
2		cotrg 6108	. . . 4 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧))
3		breq1 5151	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦◡𝐴𝑥 ↔ 𝑤◡𝐴𝑥))
4	3	anbi1d 629	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ (𝑤◡𝐴𝑥 ∧ 𝑥𝐴𝑧)))
5		breq1 5151	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑦 I 𝑧 ↔ 𝑤 I 𝑧))
6	4, 5	imbi12d 344	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑤◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑤 I 𝑧)))
7	6	albidv 1922	. . . . . 6 ⊢ (𝑦 = 𝑤 → (∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑧((𝑤◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑤 I 𝑧)))
8		breq2 5152	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑦◡𝐴𝑥 ↔ 𝑦◡𝐴𝑤))
9		breq1 5151	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥𝐴𝑧 ↔ 𝑤𝐴𝑧))
10	8, 9	anbi12d 630	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ (𝑦◡𝐴𝑤 ∧ 𝑤𝐴𝑧)))
11	10	imbi1d 341	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑦◡𝐴𝑤 ∧ 𝑤𝐴𝑧) → 𝑦 I 𝑧)))
12	11	albidv 1922	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑧((𝑦◡𝐴𝑤 ∧ 𝑤𝐴𝑧) → 𝑦 I 𝑧)))
13	7, 12	alcomw 2046	. . . . 5 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧))
14		vex 3477	. . . . . . . . 9 ⊢ 𝑦 ∈ V
15		vex 3477	. . . . . . . . 9 ⊢ 𝑥 ∈ V
16	14, 15	brcnv 5882	. . . . . . . 8 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
17	16	anbi1i 623	. . . . . . 7 ⊢ ((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧))
18		vex 3477	. . . . . . . 8 ⊢ 𝑧 ∈ V
19	18	ideq 5852	. . . . . . 7 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧)
20	17, 19	imbi12i 350	. . . . . 6 ⊢ (((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
21	20	3albii 1822	. . . . 5 ⊢ (∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
22	13, 21	bitri 275	. . . 4 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
23	2, 22	bitri 275	. . 3 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
24	23	anbi2i 622	. 2 ⊢ ((Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
25	1, 24	bitri 275	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1538   ⊆ wss 3948   class class class wbr 5148   I cid 5573  ^◡ccnv 5675   ∘ ccom 5680  Rel wrel 5681  Fun wfun 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-fun 6545

This theorem is referenced by:  dffun6  6556  dffun3OLD  6558  dffun4  6559  fundif  6597  fliftfun  7312  frrlem9  8285  fprlem1  8291  wfrlem5OLD  8319  wfrfunOLD  8325  frrlem15  9758  fpwwe2lem10  10641  fclim  15504  invfun  17718  lmfun  23205  ulmdm  26244  fundmpss  35208  fununiq  35210  fnsingle  35361  funimage  35370  funpartfun  35385