Theorem dffun2 6553

Description: Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2179, ax-12 2216. (Revised by SN, 19-Dec-2024.) Avoid ax-11 2195. (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 6116	. . . 4 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧))
3		breq1 5117	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦◡𝐴𝑥 ↔ 𝑤◡𝐴𝑥))
4	3	anbi1d 643	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ (𝑤◡𝐴𝑥 ∧ 𝑥𝐴𝑧)))
5		breq1 5117	. . . . . . . 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 5118	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑦◡𝐴𝑥 ↔ 𝑦◡𝐴𝑤))
9		breq1 5117	. . . . . . . . 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 3462	. . . . . . . . 9 ⊢ 𝑦 ∈ V
15		vex 3462	. . . . . . . . 9 ⊢ 𝑥 ∈ V
16	14, 15	brcnv 5873	. . . . . . . 8 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
17	16	anbi1i 636	. . . . . . 7 ⊢ ((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧))
18		vex 3462	. . . . . . . 8 ⊢ 𝑧 ∈ V
19	18	ideq 5843	. . . . . . 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 3908   class class class wbr 5114   I cid 5560  ^◡ccnv 5665   ∘ ccom 5670  Rel wrel 5671  Fun wfun 6537

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 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

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 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-fun 6545

This theorem is used by:  dffun6  6554  dffun4  6556  fundif  6592  fliftfun  7321  frrlem9  8300  fprlem1  8306  frrlem15  9739  fpwwe2lem10  10643  fclim  15630  invfun  17846  lmfun  23575  ulmdm  26593  fundmpss  36280  fununiq  36282  fnsingle  36430  funimage  36439  funpartfun  36456  functhincfun  50268