Theorem dff1o6 5475

Description: A one-to-one onto function in terms of function values. (Contributed by set.mm contributors, 29-Mar-2008.)

Assertion

Ref	Expression
dff1o6	⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ ran F = B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))

Distinct variable groups:   x,y,A   x,F,y

Allowed substitution hints:   B(x,y)

Proof of Theorem dff1o6

Step	Hyp	Ref	Expression
1		df-f1o 4794	. 2 ⊢ (F:A–1-1-onto→B ↔ (F:A–1-1→B ∧ F:A–onto→B))
2		dff13 5471	. . 3 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
3		df-fo 4793	. . 3 ⊢ (F:A–onto→B ↔ (F Fn A ∧ ran F = B))
4	2, 3	anbi12i 678	. 2 ⊢ ((F:A–1-1→B ∧ F:A–onto→B) ↔ ((F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)) ∧ (F Fn A ∧ ran F = B)))
5		df-3an 936	. . 3 ⊢ ((F Fn A ∧ ran F = B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)) ↔ ((F Fn A ∧ ran F = B) ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
6		eqimss 3323	. . . . . . 7 ⊢ (ran F = B → ran F ⊆ B)
7	6	anim2i 552	. . . . . 6 ⊢ ((F Fn A ∧ ran F = B) → (F Fn A ∧ ran F ⊆ B))
8		df-f 4791	. . . . . 6 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
9	7, 8	sylibr 203	. . . . 5 ⊢ ((F Fn A ∧ ran F = B) → F:A–→B)
10	9	pm4.71ri 614	. . . 4 ⊢ ((F Fn A ∧ ran F = B) ↔ (F:A–→B ∧ (F Fn A ∧ ran F = B)))
11	10	anbi1i 676	. . 3 ⊢ (((F Fn A ∧ ran F = B) ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)) ↔ ((F:A–→B ∧ (F Fn A ∧ ran F = B)) ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
12		an32 773	. . 3 ⊢ (((F:A–→B ∧ (F Fn A ∧ ran F = B)) ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)) ↔ ((F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)) ∧ (F Fn A ∧ ran F = B)))
13	5, 11, 12	3bitrri 263	. 2 ⊢ (((F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)) ∧ (F Fn A ∧ ran F = B)) ↔ (F Fn A ∧ ran F = B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
14	1, 4, 13	3bitri 262	1 ⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ ran F = B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642  ∀wral 2614   ⊆ wss 3257  ran crn 4773   Fn wfn 4776  –→wf 4777  –1-1→wf1 4778  –onto→wfo 4779  –1-1-onto→wf1o 4780   ‘cfv 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795

This theorem is referenced by:  pw1fnf1o  5855