Theorem dffun6 6586

Description: Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) Avoid ax-10 2141, ax-12 2178. (Revised by SN, 19-Dec-2024.)

Assertion

Ref	Expression
dffun6	⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))

Distinct variable group:   𝑥,𝐹,𝑦

Proof of Theorem dffun6

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun2 6583	. 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2		breq2 5170	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥𝐹𝑦 ↔ 𝑥𝐹𝑧))
3	2	mo4 2569	. . . 4 ⊢ (∃*𝑦 𝑥𝐹𝑦 ↔ ∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧))
4	3	albii 1817	. . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐹𝑦 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧))
5	4	anbi2i 622	. 2 ⊢ ((Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦) ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
6	1, 5	bitr4i 278	1 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃*wmo 2541   class class class wbr 5166  Rel wrel 5705  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-fun 6575

This theorem is referenced by:  dffun3  6587  funmo  6593  funmoOLD  6594  dffun7  6605  fununfun  6626  funcnvsn  6628  funcnv2  6646  svrelfun  6650  funimaexg  6664  fnres  6707  nfunsn  6962  dff3  7134  brdom3  10597  nqerf  10999  shftfn  15122  cnextfun  24093  perfdvf  25958  taylf  26420  funressnvmo  46960