Theorem dffun6 6549

Description: Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) Avoid ax-10 2178, ax-12 2213. (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 6548	. 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2		breq2 5107	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥𝐹𝑦 ↔ 𝑥𝐹𝑧))
3	2	mo4 2592	. . . 4 ⊢ (∃*𝑦 𝑥𝐹𝑦 ↔ ∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧))
4	3	albii 1852	. . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐹𝑦 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧))
5	4	anbi2i 635	. 2 ⊢ ((Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦) ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
6	1, 5	bitr4i 281	1 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃*wmo 2563   class class class wbr 5103  Rel wrel 5656  Fun wfun 6532

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 2733  ax-sep 5249  ax-pr 5391

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-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-fun 6540

This theorem is used by:  dffun3  6550  funmo  6554  dffun7  6567  fununfun  6588  funcnvsn  6590  funcnv2  6608  svrelfun  6612  funimaexg  6626  fnres  6666  nfunsn  6924  dff3  7100  brdom3  10607  nqerf  11015  shftfn  15226  cnextfun  24383  perfdvf  26223  taylf  26688  funressnvmo  48114