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Theorem dff13 5763
Description: A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.)
Assertion
Ref Expression
dff13 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((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 5416 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑧∃*𝑥 𝑥𝐹𝑧))
2 ffn 5361 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 vex 2740 . . . . . . . . . . . . . . 15 𝑥 ∈ V
4 vex 2740 . . . . . . . . . . . . . . 15 𝑧 ∈ V
53, 4breldm 4827 . . . . . . . . . . . . . 14 (𝑥𝐹𝑧𝑥 ∈ dom 𝐹)
6 fndm 5311 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76eleq2d 2247 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → (𝑥 ∈ dom 𝐹𝑥𝐴))
85, 7imbitrid 154 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 → (𝑥𝐹𝑧𝑥𝐴))
9 vex 2740 . . . . . . . . . . . . . . 15 𝑦 ∈ V
109, 4breldm 4827 . . . . . . . . . . . . . 14 (𝑦𝐹𝑧𝑦 ∈ dom 𝐹)
116eleq2d 2247 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → (𝑦 ∈ dom 𝐹𝑦𝐴))
1210, 11imbitrid 154 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 → (𝑦𝐹𝑧𝑦𝐴))
138, 12anim12d 335 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧𝑦𝐹𝑧) → (𝑥𝐴𝑦𝐴)))
1413pm4.71rd 394 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧𝑦𝐹𝑧) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝑥𝐹𝑧𝑦𝐹𝑧))))
15 eqcom 2179 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑧)
16 fnbrfvb 5552 . . . . . . . . . . . . . . 15 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑧𝑥𝐹𝑧))
1715, 16bitrid 192 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑧 = (𝐹𝑥) ↔ 𝑥𝐹𝑧))
18 eqcom 2179 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑦) ↔ (𝐹𝑦) = 𝑧)
19 fnbrfvb 5552 . . . . . . . . . . . . . . 15 ((𝐹 Fn 𝐴𝑦𝐴) → ((𝐹𝑦) = 𝑧𝑦𝐹𝑧))
2018, 19bitrid 192 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴𝑦𝐴) → (𝑧 = (𝐹𝑦) ↔ 𝑦𝐹𝑧))
2117, 20bi2anan9 606 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴𝑥𝐴) ∧ (𝐹 Fn 𝐴𝑦𝐴)) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) ↔ (𝑥𝐹𝑧𝑦𝐹𝑧)))
2221anandis 592 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) ↔ (𝑥𝐹𝑧𝑦𝐹𝑧)))
2322pm5.32da 452 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → (((𝑥𝐴𝑦𝐴) ∧ (𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦))) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝑥𝐹𝑧𝑦𝐹𝑧))))
2414, 23bitr4d 191 . . . . . . . . . 10 (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧𝑦𝐹𝑧) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)))))
2524imbi1d 231 . . . . . . . . 9 (𝐹 Fn 𝐴 → (((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ (((𝑥𝐴𝑦𝐴) ∧ (𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦))) → 𝑥 = 𝑦)))
26 impexp 263 . . . . . . . . 9 ((((𝑥𝐴𝑦𝐴) ∧ (𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦))) → 𝑥 = 𝑦) ↔ ((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)))
2725, 26bitrdi 196 . . . . . . . 8 (𝐹 Fn 𝐴 → (((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦))))
2827albidv 1824 . . . . . . 7 (𝐹 Fn 𝐴 → (∀𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑧((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦))))
29 19.21v 1873 . . . . . . . 8 (∀𝑧((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥𝐴𝑦𝐴) → ∀𝑧((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)))
30 19.23v 1883 . . . . . . . . . . 11 (∀𝑧((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦))
31 funfvex 5528 . . . . . . . . . . . . . 14 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
3231funfni 5312 . . . . . . . . . . . . 13 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ V)
33 eqvincg 2861 . . . . . . . . . . . . 13 ((𝐹𝑥) ∈ V → ((𝐹𝑥) = (𝐹𝑦) ↔ ∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦))))
3432, 33syl 14 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = (𝐹𝑦) ↔ ∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦))))
3534imbi1d 231 . . . . . . . . . . 11 ((𝐹 Fn 𝐴𝑥𝐴) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)))
3630, 35bitr4id 199 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑥𝐴) → (∀𝑧((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3736adantrr 479 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (∀𝑧((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3837pm5.74da 443 . . . . . . . 8 (𝐹 Fn 𝐴 → (((𝑥𝐴𝑦𝐴) → ∀𝑧((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
3929, 38bitrid 192 . . . . . . 7 (𝐹 Fn 𝐴 → (∀𝑧((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
4028, 39bitrd 188 . . . . . 6 (𝐹 Fn 𝐴 → (∀𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
41402albidv 1867 . . . . 5 (𝐹 Fn 𝐴 → (∀𝑥𝑦𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
42 breq1 4003 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐹𝑧𝑦𝐹𝑧))
4342mo4 2087 . . . . . . 7 (∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥𝑦((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦))
4443albii 1470 . . . . . 6 (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑧𝑥𝑦((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦))
45 alrot3 1485 . . . . . 6 (∀𝑧𝑥𝑦((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦))
4644, 45bitri 184 . . . . 5 (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥𝑦𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦))
47 r2al 2496 . . . . 5 (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
4841, 46, 473bitr4g 223 . . . 4 (𝐹 Fn 𝐴 → (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
492, 48syl 14 . . 3 (𝐹:𝐴𝐵 → (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
5049pm5.32i 454 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑧∃*𝑥 𝑥𝐹𝑧) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
511, 50bitri 184 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  wex 1492  ∃*wmo 2027  wcel 2148  wral 2455  Vcvv 2737   class class class wbr 4000  dom cdm 4623   Fn wfn 5207  wf 5208  1-1wf1 5209  cfv 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fv 5220
This theorem is referenced by:  f1veqaeq  5764  dff13f  5765  dff1o6  5771  fcof1  5778  f1o2ndf1  6223  cc2lem  7256  cnref1o  9639  frec2uzf1od  10392  iseqf1olemqf1o  10479  reeff1  11692  crth  12207  eulerthlemh  12214  1arith  12348  nninfdclemf1  12436  ioocosf1o  13942  peano4nninf  14411  exmidsbthrlem  14426
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