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Theorem dff13f 5472
 Description: A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)
Hypotheses
Ref Expression
dff13f.1 xF
dff13f.2 yF
Assertion
Ref Expression
dff13f (F:A1-1B ↔ (F:A–→B x A y A ((Fx) = (Fy) → x = y)))
Distinct variable group:   x,y,A
Allowed substitution hints:   B(x,y)   F(x,y)

Proof of Theorem dff13f
Dummy variables w v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dff13 5471 . 2 (F:A1-1B ↔ (F:A–→B w A v A ((Fw) = (Fv) → w = v)))
2 dff13f.2 . . . . . . . . 9 yF
3 nfcv 2489 . . . . . . . . 9 yw
42, 3nffv 5334 . . . . . . . 8 y(Fw)
5 nfcv 2489 . . . . . . . . 9 yv
62, 5nffv 5334 . . . . . . . 8 y(Fv)
74, 6nfeq 2496 . . . . . . 7 y(Fw) = (Fv)
8 nfv 1619 . . . . . . 7 y w = v
97, 8nfim 1813 . . . . . 6 y((Fw) = (Fv) → w = v)
10 nfv 1619 . . . . . 6 v((Fw) = (Fy) → w = y)
11 fveq2 5328 . . . . . . . 8 (v = y → (Fv) = (Fy))
1211eqeq2d 2364 . . . . . . 7 (v = y → ((Fw) = (Fv) ↔ (Fw) = (Fy)))
13 eqeq2 2362 . . . . . . 7 (v = y → (w = vw = y))
1412, 13imbi12d 311 . . . . . 6 (v = y → (((Fw) = (Fv) → w = v) ↔ ((Fw) = (Fy) → w = y)))
159, 10, 14cbvral 2831 . . . . 5 (v A ((Fw) = (Fv) → w = v) ↔ y A ((Fw) = (Fy) → w = y))
1615ralbii 2638 . . . 4 (w A v A ((Fw) = (Fv) → w = v) ↔ w A y A ((Fw) = (Fy) → w = y))
17 nfcv 2489 . . . . . 6 xA
18 dff13f.1 . . . . . . . . 9 xF
19 nfcv 2489 . . . . . . . . 9 xw
2018, 19nffv 5334 . . . . . . . 8 x(Fw)
21 nfcv 2489 . . . . . . . . 9 xy
2218, 21nffv 5334 . . . . . . . 8 x(Fy)
2320, 22nfeq 2496 . . . . . . 7 x(Fw) = (Fy)
24 nfv 1619 . . . . . . 7 x w = y
2523, 24nfim 1813 . . . . . 6 x((Fw) = (Fy) → w = y)
2617, 25nfral 2667 . . . . 5 xy A ((Fw) = (Fy) → w = y)
27 nfv 1619 . . . . 5 wy A ((Fx) = (Fy) → x = y)
28 fveq2 5328 . . . . . . . 8 (w = x → (Fw) = (Fx))
2928eqeq1d 2361 . . . . . . 7 (w = x → ((Fw) = (Fy) ↔ (Fx) = (Fy)))
30 eqeq1 2359 . . . . . . 7 (w = x → (w = yx = y))
3129, 30imbi12d 311 . . . . . 6 (w = x → (((Fw) = (Fy) → w = y) ↔ ((Fx) = (Fy) → x = y)))
3231ralbidv 2634 . . . . 5 (w = x → (y A ((Fw) = (Fy) → w = y) ↔ y A ((Fx) = (Fy) → x = y)))
3326, 27, 32cbvral 2831 . . . 4 (w A y A ((Fw) = (Fy) → w = y) ↔ x A y A ((Fx) = (Fy) → x = y))
3416, 33bitri 240 . . 3 (w A v A ((Fw) = (Fv) → w = v) ↔ x A y A ((Fx) = (Fy) → x = y))
3534anbi2i 675 . 2 ((F:A–→B w A v A ((Fw) = (Fv) → w = v)) ↔ (F:A–→B x A y A ((Fx) = (Fy) → x = y)))
361, 35bitri 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   = wceq 1642  Ⅎwnfc 2476  ∀wral 2614  –→wf 4777  –1-1→wf1 4778   ‘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-fv 4795 This theorem is referenced by: (None)
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