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Theorem dff13f 5671
 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:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((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 5669 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣)))
2 dff13f.2 . . . . . . . . 9 𝑦𝐹
3 nfcv 2281 . . . . . . . . 9 𝑦𝑤
42, 3nffv 5431 . . . . . . . 8 𝑦(𝐹𝑤)
5 nfcv 2281 . . . . . . . . 9 𝑦𝑣
62, 5nffv 5431 . . . . . . . 8 𝑦(𝐹𝑣)
74, 6nfeq 2289 . . . . . . 7 𝑦(𝐹𝑤) = (𝐹𝑣)
8 nfv 1508 . . . . . . 7 𝑦 𝑤 = 𝑣
97, 8nfim 1551 . . . . . 6 𝑦((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣)
10 nfv 1508 . . . . . 6 𝑣((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)
11 fveq2 5421 . . . . . . . 8 (𝑣 = 𝑦 → (𝐹𝑣) = (𝐹𝑦))
1211eqeq2d 2151 . . . . . . 7 (𝑣 = 𝑦 → ((𝐹𝑤) = (𝐹𝑣) ↔ (𝐹𝑤) = (𝐹𝑦)))
13 equequ2 1689 . . . . . . 7 (𝑣 = 𝑦 → (𝑤 = 𝑣𝑤 = 𝑦))
1412, 13imbi12d 233 . . . . . 6 (𝑣 = 𝑦 → (((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)))
159, 10, 14cbvral 2650 . . . . 5 (∀𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ∀𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦))
1615ralbii 2441 . . . 4 (∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ∀𝑤𝐴𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦))
17 nfcv 2281 . . . . . 6 𝑥𝐴
18 dff13f.1 . . . . . . . . 9 𝑥𝐹
19 nfcv 2281 . . . . . . . . 9 𝑥𝑤
2018, 19nffv 5431 . . . . . . . 8 𝑥(𝐹𝑤)
21 nfcv 2281 . . . . . . . . 9 𝑥𝑦
2218, 21nffv 5431 . . . . . . . 8 𝑥(𝐹𝑦)
2320, 22nfeq 2289 . . . . . . 7 𝑥(𝐹𝑤) = (𝐹𝑦)
24 nfv 1508 . . . . . . 7 𝑥 𝑤 = 𝑦
2523, 24nfim 1551 . . . . . 6 𝑥((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)
2617, 25nfralxy 2471 . . . . 5 𝑥𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)
27 nfv 1508 . . . . 5 𝑤𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)
28 fveq2 5421 . . . . . . . 8 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
2928eqeq1d 2148 . . . . . . 7 (𝑤 = 𝑥 → ((𝐹𝑤) = (𝐹𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
30 equequ1 1688 . . . . . . 7 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
3129, 30imbi12d 233 . . . . . 6 (𝑤 = 𝑥 → (((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3231ralbidv 2437 . . . . 5 (𝑤 = 𝑥 → (∀𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦) ↔ ∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3326, 27, 32cbvral 2650 . . . 4 (∀𝑤𝐴𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3416, 33bitri 183 . . 3 (∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3534anbi2i 452 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣)) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
361, 35bitri 183 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  Ⅎwnfc 2268  ∀wral 2416  ⟶wf 5119  –1-1→wf1 5120  ‘cfv 5123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fv 5131 This theorem is referenced by:  f1mpt  5672  dom2lem  6666
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