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Theorem dff13f 6704
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 6703 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣)))
2 dff13f.2 . . . . . . . . 9 𝑦𝐹
3 nfcv 2906 . . . . . . . . 9 𝑦𝑤
42, 3nffv 6384 . . . . . . . 8 𝑦(𝐹𝑤)
5 nfcv 2906 . . . . . . . . 9 𝑦𝑣
62, 5nffv 6384 . . . . . . . 8 𝑦(𝐹𝑣)
74, 6nfeq 2918 . . . . . . 7 𝑦(𝐹𝑤) = (𝐹𝑣)
8 nfv 2009 . . . . . . 7 𝑦 𝑤 = 𝑣
97, 8nfim 1995 . . . . . 6 𝑦((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣)
10 nfv 2009 . . . . . 6 𝑣((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)
11 fveq2 6374 . . . . . . . 8 (𝑣 = 𝑦 → (𝐹𝑣) = (𝐹𝑦))
1211eqeq2d 2774 . . . . . . 7 (𝑣 = 𝑦 → ((𝐹𝑤) = (𝐹𝑣) ↔ (𝐹𝑤) = (𝐹𝑦)))
13 equequ2 2123 . . . . . . 7 (𝑣 = 𝑦 → (𝑤 = 𝑣𝑤 = 𝑦))
1412, 13imbi12d 335 . . . . . 6 (𝑣 = 𝑦 → (((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)))
159, 10, 14cbvral 3314 . . . . 5 (∀𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ∀𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦))
1615ralbii 3126 . . . 4 (∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ∀𝑤𝐴𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦))
17 nfcv 2906 . . . . . 6 𝑥𝐴
18 dff13f.1 . . . . . . . . 9 𝑥𝐹
19 nfcv 2906 . . . . . . . . 9 𝑥𝑤
2018, 19nffv 6384 . . . . . . . 8 𝑥(𝐹𝑤)
21 nfcv 2906 . . . . . . . . 9 𝑥𝑦
2218, 21nffv 6384 . . . . . . . 8 𝑥(𝐹𝑦)
2320, 22nfeq 2918 . . . . . . 7 𝑥(𝐹𝑤) = (𝐹𝑦)
24 nfv 2009 . . . . . . 7 𝑥 𝑤 = 𝑦
2523, 24nfim 1995 . . . . . 6 𝑥((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)
2617, 25nfral 3091 . . . . 5 𝑥𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)
27 nfv 2009 . . . . 5 𝑤𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)
28 fveqeq2 6383 . . . . . . 7 (𝑤 = 𝑥 → ((𝐹𝑤) = (𝐹𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
29 equequ1 2122 . . . . . . 7 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
3028, 29imbi12d 335 . . . . . 6 (𝑤 = 𝑥 → (((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3130ralbidv 3132 . . . . 5 (𝑤 = 𝑥 → (∀𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦) ↔ ∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3226, 27, 31cbvral 3314 . . . 4 (∀𝑤𝐴𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3316, 32bitri 266 . . 3 (∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3433anbi2i 616 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣)) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
351, 34bitri 266 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wnfc 2893  wral 3054  wf 6063  1-1wf1 6064  cfv 6067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-br 4809  df-opab 4871  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fv 6075
This theorem is referenced by:  f1mpt  6709  dom2lem  8199
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