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Theorem dff13 5471
 Description: A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by set.mm contributors, 29-Oct-1996.)
Assertion
Ref Expression
dff13 (F:A1-1B ↔ (F:A–→B x A y A ((Fx) = (Fy) → x = y)))
Distinct variable groups:   x,y,A   x,F,y
Allowed substitution hints:   B(x,y)

Proof of Theorem dff13
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 dff12 5257 . 2 (F:A1-1B ↔ (F:A–→B z∃*x xFz))
2 ffn 5223 . . . 4 (F:A–→BF Fn A)
3 breldm 4911 . . . . . . . . . . . . . 14 (xFzx dom F)
4 fndm 5182 . . . . . . . . . . . . . . 15 (F Fn A → dom F = A)
54eleq2d 2420 . . . . . . . . . . . . . 14 (F Fn A → (x dom Fx A))
63, 5syl5ib 210 . . . . . . . . . . . . 13 (F Fn A → (xFzx A))
7 breldm 4911 . . . . . . . . . . . . . 14 (yFzy dom F)
84eleq2d 2420 . . . . . . . . . . . . . 14 (F Fn A → (y dom Fy A))
97, 8syl5ib 210 . . . . . . . . . . . . 13 (F Fn A → (yFzy A))
106, 9anim12d 546 . . . . . . . . . . . 12 (F Fn A → ((xFz yFz) → (x A y A)))
1110pm4.71rd 616 . . . . . . . . . . 11 (F Fn A → ((xFz yFz) ↔ ((x A y A) (xFz yFz))))
12 eqcom 2355 . . . . . . . . . . . . . . 15 (z = (Fx) ↔ (Fx) = z)
13 fnbrfvb 5358 . . . . . . . . . . . . . . 15 ((F Fn A x A) → ((Fx) = zxFz))
1412, 13syl5bb 248 . . . . . . . . . . . . . 14 ((F Fn A x A) → (z = (Fx) ↔ xFz))
15 eqcom 2355 . . . . . . . . . . . . . . 15 (z = (Fy) ↔ (Fy) = z)
16 fnbrfvb 5358 . . . . . . . . . . . . . . 15 ((F Fn A y A) → ((Fy) = zyFz))
1715, 16syl5bb 248 . . . . . . . . . . . . . 14 ((F Fn A y A) → (z = (Fy) ↔ yFz))
1814, 17bi2anan9 843 . . . . . . . . . . . . 13 (((F Fn A x A) (F Fn A y A)) → ((z = (Fx) z = (Fy)) ↔ (xFz yFz)))
1918anandis 803 . . . . . . . . . . . 12 ((F Fn A (x A y A)) → ((z = (Fx) z = (Fy)) ↔ (xFz yFz)))
2019pm5.32da 622 . . . . . . . . . . 11 (F Fn A → (((x A y A) (z = (Fx) z = (Fy))) ↔ ((x A y A) (xFz yFz))))
2111, 20bitr4d 247 . . . . . . . . . 10 (F Fn A → ((xFz yFz) ↔ ((x A y A) (z = (Fx) z = (Fy)))))
2221imbi1d 308 . . . . . . . . 9 (F Fn A → (((xFz yFz) → x = y) ↔ (((x A y A) (z = (Fx) z = (Fy))) → x = y)))
23 impexp 433 . . . . . . . . 9 ((((x A y A) (z = (Fx) z = (Fy))) → x = y) ↔ ((x A y A) → ((z = (Fx) z = (Fy)) → x = y)))
2422, 23syl6bb 252 . . . . . . . 8 (F Fn A → (((xFz yFz) → x = y) ↔ ((x A y A) → ((z = (Fx) z = (Fy)) → x = y))))
2524albidv 1625 . . . . . . 7 (F Fn A → (z((xFz yFz) → x = y) ↔ z((x A y A) → ((z = (Fx) z = (Fy)) → x = y))))
26 19.21v 1890 . . . . . . . 8 (z((x A y A) → ((z = (Fx) z = (Fy)) → x = y)) ↔ ((x A y A) → z((z = (Fx) z = (Fy)) → x = y)))
27 19.23v 1891 . . . . . . . . . 10 (z((z = (Fx) z = (Fy)) → x = y) ↔ (z(z = (Fx) z = (Fy)) → x = y))
28 fvex 5339 . . . . . . . . . . . 12 (Fx) V
2928eqvinc 2966 . . . . . . . . . . 11 ((Fx) = (Fy) ↔ z(z = (Fx) z = (Fy)))
3029imbi1i 315 . . . . . . . . . 10 (((Fx) = (Fy) → x = y) ↔ (z(z = (Fx) z = (Fy)) → x = y))
3127, 30bitr4i 243 . . . . . . . . 9 (z((z = (Fx) z = (Fy)) → x = y) ↔ ((Fx) = (Fy) → x = y))
3231imbi2i 303 . . . . . . . 8 (((x A y A) → z((z = (Fx) z = (Fy)) → x = y)) ↔ ((x A y A) → ((Fx) = (Fy) → x = y)))
3326, 32bitri 240 . . . . . . 7 (z((x A y A) → ((z = (Fx) z = (Fy)) → x = y)) ↔ ((x A y A) → ((Fx) = (Fy) → x = y)))
3425, 33syl6bb 252 . . . . . 6 (F Fn A → (z((xFz yFz) → x = y) ↔ ((x A y A) → ((Fx) = (Fy) → x = y))))
35342albidv 1627 . . . . 5 (F Fn A → (xyz((xFz yFz) → x = y) ↔ xy((x A y A) → ((Fx) = (Fy) → x = y))))
36 breq1 4642 . . . . . . . 8 (x = y → (xFzyFz))
3736mo4 2237 . . . . . . 7 (∃*x xFzxy((xFz yFz) → x = y))
3837albii 1566 . . . . . 6 (z∃*x xFzzxy((xFz yFz) → x = y))
39 alcom 1737 . . . . . 6 (zxy((xFz yFz) → x = y) ↔ xzy((xFz yFz) → x = y))
40 alcom 1737 . . . . . . 7 (zy((xFz yFz) → x = y) ↔ yz((xFz yFz) → x = y))
4140albii 1566 . . . . . 6 (xzy((xFz yFz) → x = y) ↔ xyz((xFz yFz) → x = y))
4238, 39, 413bitri 262 . . . . 5 (z∃*x xFzxyz((xFz yFz) → x = y))
43 r2al 2651 . . . . 5 (x A y A ((Fx) = (Fy) → x = y) ↔ xy((x A y A) → ((Fx) = (Fy) → x = y)))
4435, 42, 433bitr4g 279 . . . 4 (F Fn A → (z∃*x xFzx A y A ((Fx) = (Fy) → x = y)))
452, 44syl 15 . . 3 (F:A–→B → (z∃*x xFzx A y A ((Fx) = (Fy) → x = y)))
4645pm5.32i 618 . 2 ((F:A–→B z∃*x xFz) ↔ (F:A–→B x A y A ((Fx) = (Fy) → x = y)))
471, 46bitri 240 1 (F:A1-1B ↔ (F:A–→B x A y A ((Fx) = (Fy) → x = y)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ∀wral 2614   class class class wbr 4639  dom cdm 4772   Fn wfn 4776  –→wf 4777  –1-1→wf1 4778   ‘cfv 4781 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-fv 4795 This theorem is referenced by:  dff13f  5472  f1fveq  5473  dff1o6  5475
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