Theorem f1cnv 2782

Description: Two ways of expressing that a set A (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un₂ (A)" for one-to-one. We do not introduce a separate notation since we do not use it very often.

Assertion

Ref	Expression
f1cnv	⊢ (◡◡A:dom A–1-1→V ↔ (Fun ◡A ∧ Fun ◡◡A))

Proof of Theorem f1cnv

Step	Hyp	Ref	Expression
1		df-f1 2435	. 2 ⊢ (◡◡A:dom A–1-1→V ↔ (◡◡A:dom A–→V ∧ Fun ◡◡◡A))
2		fnf 2753	. . . 4 ⊢ (◡◡A Fn dom A ↔ ◡◡A:dom A–→V)
3		df-fn 2433	. . . . 5 ⊢ (◡◡A Fn dom A ↔ (Fun ◡◡A ∧ dom ◡◡A = dom A))
4		dfdm4 2525	. . . . . 6 ⊢ dom A = ran ◡A
5		df-rn 2429	. . . . . 6 ⊢ ran ◡A = dom ◡◡A
6	4, 5	eqtr2 1120	. . . . 5 ⊢ dom ◡◡A = dom A
7	3, 6	mpbiranr 548	. . . 4 ⊢ (◡◡A Fn dom A ↔ Fun ◡◡A)
8	2, 7	bitr3 153	. . 3 ⊢ (◡◡A:dom A–→V ↔ Fun ◡◡A)
9		relcnv 2624	. . . . 5 ⊢ Rel ◡A
10		dfrel2 2660	. . . . 5 ⊢ (Rel ◡A ↔ ◡◡◡A = ◡A)
11	9, 10	mpbi 164	. . . 4 ⊢ ◡◡◡A = ◡A
12		funeq 2683	. . . 4 ⊢ (◡◡◡A = ◡A → (Fun ◡◡◡A ↔ Fun ◡A))
13	11, 12	ax-mp 6	. . 3 ⊢ (Fun ◡◡◡A ↔ Fun ◡A)
14	8, 13	anbi12i 369	. 2 ⊢ ((◡◡A:dom A–→V ∧ Fun ◡◡◡A) ↔ (Fun ◡◡A ∧ Fun ◡A))
15		ancom 333	. 2 ⊢ ((Fun ◡◡A ∧ Fun ◡A) ↔ (Fun ◡A ∧ Fun ◡◡A))
16	1, 14, 15	3bitr 155	1 ⊢ (◡◡A:dom A–1-1→V ↔ (Fun ◡A ∧ Fun ◡◡A))

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091  Vcvv 1348  ^◡ccnv 2409  dom cdm 2410  ran crn 2411  Rel wrel 2415  Fun wfun 2416   Fn wfn 2417  –→wf 2418  –1-1→wf1 2419

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-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435

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