Theorem dffun2 6573

Description: Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2139, ax-12 2175. (Revised by SN, 19-Dec-2024.) Avoid ax-11 2155. (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 6565	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))
2		cotrg 6130	. . . 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 631	. . . . . . . 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 1918	. . . . . 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 632	. . . . . . . 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 1918	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑧((𝑦◡𝐴𝑤 ∧ 𝑤𝐴𝑧) → 𝑦 I 𝑧)))
13	7, 12	alcomw 2042	. . . . 5 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 I 𝑧))
14		vex 3482	. . . . . . . . 9 ⊢ 𝑦 ∈ V
15		vex 3482	. . . . . . . . 9 ⊢ 𝑥 ∈ V
16	14, 15	brcnv 5896	. . . . . . . 8 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
17	16	anbi1i 624	. . . . . . 7 ⊢ ((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧))
18		vex 3482	. . . . . . . 8 ⊢ 𝑧 ∈ V
19	18	ideq 5866	. . . . . . 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 1818	. . . . 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 623	. 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 206   ∧ wa 395  ∀wal 1535   ⊆ wss 3963   class class class wbr 5148   I cid 5582  ^◡ccnv 5688   ∘ ccom 5693  Rel wrel 5694  Fun wfun 6557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-fun 6565

This theorem is referenced by:  dffun6  6576  dffun3OLD  6578  dffun4  6579  fundif  6617  fliftfun  7332  frrlem9  8318  fprlem1  8324  wfrlem5OLD  8352  wfrfunOLD  8358  frrlem15  9795  fpwwe2lem10  10678  fclim  15586  invfun  17812  lmfun  23405  ulmdm  26451  fundmpss  35748  fununiq  35750  fnsingle  35901  funimage  35910  funpartfun  35925