Theorem f1ocnv 2811

Description: The converse of a one-to-one onto function is also one-to-one onto.

Assertion

Ref	Expression
f1ocnv	⊢ (F:A–1-1-onto→B → ◡F:B–1-1-onto→A)

Proof of Theorem f1ocnv

Step	Hyp	Ref	Expression
1		df-rn 2429	. . . . . . . 8 ⊢ ran F = dom ◡F
2	1	cleq1i 1108	. . . . . . 7 ⊢ (ran F = B ↔ dom ◡F = B)
3	2	anbi2i 367	. . . . . 6 ⊢ ((Fun ◡F ∧ ran F = B) ↔ (Fun ◡F ∧ dom ◡F = B))
4		df-fn 2433	. . . . . 6 ⊢ (◡F Fn B ↔ (Fun ◡F ∧ dom ◡F = B))
5	3, 4	bitr4 154	. . . . 5 ⊢ ((Fun ◡F ∧ ran F = B) ↔ ◡F Fn B)
6	5	biimp 133	. . . 4 ⊢ ((Fun ◡F ∧ ran F = B) → ◡F Fn B)
7		fnfun 2721	. . . . . 6 ⊢ (F Fn A → Fun F)
8		funcnvcnv 2701	. . . . . 6 ⊢ (Fun F → Fun ◡◡F)
9	7, 8	syl 12	. . . . 5 ⊢ (F Fn A → Fun ◡◡F)
10		fndm 2723	. . . . . 6 ⊢ (F Fn A → dom F = A)
11		dfdm4 2525	. . . . . 6 ⊢ dom F = ran ◡F
12	10, 11	syl5eqr 1138	. . . . 5 ⊢ (F Fn A → ran ◡F = A)
13	9, 12	jca 236	. . . 4 ⊢ (F Fn A → (Fun ◡◡F ∧ ran ◡F = A))
14	6, 13	anim12i 268	. . 3 ⊢ (((Fun ◡F ∧ ran F = B) ∧ F Fn A) → (◡F Fn B ∧ (Fun ◡◡F ∧ ran ◡F = A)))
15	14	ancoms 334	. 2 ⊢ ((F Fn A ∧ (Fun ◡F ∧ ran F = B)) → (◡F Fn B ∧ (Fun ◡◡F ∧ ran ◡F = A)))
16		f1o2 2804	. . 3 ⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ Fun ◡F ∧ ran F = B))
17		3anass 585	. . 3 ⊢ ((F Fn A ∧ Fun ◡F ∧ ran F = B) ↔ (F Fn A ∧ (Fun ◡F ∧ ran F = B)))
18	16, 17	bitr 151	. 2 ⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ (Fun ◡F ∧ ran F = B)))
19		f1o2 2804	. . 3 ⊢ (◡F:B–1-1-onto→A ↔ (◡F Fn B ∧ Fun ◡◡F ∧ ran ◡F = A))
20		3anass 585	. . 3 ⊢ ((◡F Fn B ∧ Fun ◡◡F ∧ ran ◡F = A) ↔ (◡F Fn B ∧ (Fun ◡◡F ∧ ran ◡F = A)))
21	19, 20	bitr 151	. 2 ⊢ (◡F:B–1-1-onto→A ↔ (◡F Fn B ∧ (Fun ◡◡F ∧ ran ◡F = A)))
22	15, 18, 21	3imtr4 192	1 ⊢ (F:A–1-1-onto→B → ◡F:B–1-1-onto→A)

Syntax hints:   → wi 2   ∧ wa 196   ∧ w3a 581   = wceq 1091  ^◡ccnv 2409  dom cdm 2410  ran crn 2411  Fun wfun 2416   Fn wfn 2417  –1-1-onto→wf1o 2421

This theorem is referenced by:  f1ocnvb 2812  f1imacnv 2814  f1ococnv2 2817  f1ococnv1 2818  f1dmex 2819  f1ocnvfv1 2919  f1ocnvfv2 2920  isocnv 2934  ener 3313  en0 3328  en1 3331  mapenlem2 3385  ssenen 3399  weth 3602  uzrdgval 4657  uzrdgsuc 4659

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-3an 583  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  df-fo 2436  df-f1o 2437

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