Theorem dff12 6775

Description: Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)

Assertion

Ref	Expression
dff12	⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))

Distinct variable group:   𝑥,𝑦,𝐹

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem dff12

Step	Hyp	Ref	Expression
1		df-f1 6543	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
2		funcnv2 6606	. . 3 ⊢ (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
3	2	anbi2i 634	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))
4	1, 3	bitri 278	1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∀wal 1568  ∃*wmo 2565   class class class wbr 5110  ^◡ccnv 5662  Fun wfun 6532  ⟶wf 6534  –1-1→wf1 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-fun 6540  df-f1 6543

This theorem is referenced by:  dff13  7254  fseqenlem2  10010  s4f1o  14957  2ndcdisj  23594  usgrexmplef  29587  phpreu  38233  sinnpoly  47605