Theorem dffun6 6577

Description: Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) Avoid ax-10 2180, ax-12 2216. (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 6576	. 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2		breq2 5112	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥𝐹𝑦 ↔ 𝑥𝐹𝑧))
3	2	mo4 2594	. . . 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 2565   class class class wbr 5108  Rel wrel 5663  Fun wfun 6560

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-fun 6568

This theorem is used by:  dffun3  6578  funmo  6582  dffun7  6595  fununfunOLD  6617  funcnvsn  6620  funcnv2  6638  svrelfun  6642  funimaexg  6656  fnres  6696  nfunsn  6955  dff3  7132  brdom3  10646  nqerf  11054  shftfn  15269  cnextfun  24438  perfdvf  26278  taylf  26745  funressnvmo  48388