Theorem f11 2780

Description: Alternate definition of a one-to-one function.

Assertion

Ref	Expression
f11	⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀y∃*x xFy))

Distinct variable group(s):   x,y,F

Proof of Theorem f11

Step	Hyp	Ref	Expression
1		df-f1 2435	. 2 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F))
2		dffunmo 2679	. . . . 5 ⊢ (Fun ◡F ↔ (Rel ◡F ∧ ∀y∃*x y◡Fx))
3		relcnv 2624	. . . . 5 ⊢ Rel ◡F
4	2, 3	mpbiran 547	. . . 4 ⊢ (Fun ◡F ↔ ∀y∃*x y◡Fx)
5		visset 1350	. . . . . . 7 ⊢ y ∈ V
6		visset 1350	. . . . . . 7 ⊢ x ∈ V
7	5, 6	brcnv 2519	. . . . . 6 ⊢ (y◡Fx ↔ xFy)
8	7	bimo 1031	. . . . 5 ⊢ (∃*x y◡Fx ↔ ∃*x xFy)
9	8	bial 695	. . . 4 ⊢ (∀y∃*x y◡Fx ↔ ∀y∃*x xFy)
10	4, 9	bitr 151	. . 3 ⊢ (Fun ◡F ↔ ∀y∃*x xFy)
11	10	anbi2i 367	. 2 ⊢ ((F:A–→B ∧ Fun ◡F) ↔ (F:A–→B ∧ ∀y∃*x xFy))
12	1, 11	bitr 151	1 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀y∃*x xFy))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃*wmo 1008   class class class wbr 2054  ^◡ccnv 2409  Rel wrel 2415  Fun wfun 2416  –→wf 2418  –1-1→wf1 2419

This theorem is referenced by:  f1fv 2916

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-fun 2432  df-f1 2435

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