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Theorem dff13f 7266
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 7265 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣)))
2 dff13f.2 . . . . . . . . 9 𝑦𝐹
3 nfcv 2899 . . . . . . . . 9 𝑦𝑤
42, 3nffv 6907 . . . . . . . 8 𝑦(𝐹𝑤)
5 nfcv 2899 . . . . . . . . 9 𝑦𝑣
62, 5nffv 6907 . . . . . . . 8 𝑦(𝐹𝑣)
74, 6nfeq 2913 . . . . . . 7 𝑦(𝐹𝑤) = (𝐹𝑣)
8 nfv 1910 . . . . . . 7 𝑦 𝑤 = 𝑣
97, 8nfim 1892 . . . . . 6 𝑦((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣)
10 nfv 1910 . . . . . 6 𝑣((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)
11 fveq2 6897 . . . . . . . 8 (𝑣 = 𝑦 → (𝐹𝑣) = (𝐹𝑦))
1211eqeq2d 2739 . . . . . . 7 (𝑣 = 𝑦 → ((𝐹𝑤) = (𝐹𝑣) ↔ (𝐹𝑤) = (𝐹𝑦)))
13 equequ2 2022 . . . . . . 7 (𝑣 = 𝑦 → (𝑤 = 𝑣𝑤 = 𝑦))
1412, 13imbi12d 344 . . . . . 6 (𝑣 = 𝑦 → (((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)))
159, 10, 14cbvralw 3300 . . . . 5 (∀𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ∀𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦))
1615ralbii 3090 . . . 4 (∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ∀𝑤𝐴𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦))
17 nfcv 2899 . . . . . 6 𝑥𝐴
18 dff13f.1 . . . . . . . . 9 𝑥𝐹
19 nfcv 2899 . . . . . . . . 9 𝑥𝑤
2018, 19nffv 6907 . . . . . . . 8 𝑥(𝐹𝑤)
21 nfcv 2899 . . . . . . . . 9 𝑥𝑦
2218, 21nffv 6907 . . . . . . . 8 𝑥(𝐹𝑦)
2320, 22nfeq 2913 . . . . . . 7 𝑥(𝐹𝑤) = (𝐹𝑦)
24 nfv 1910 . . . . . . 7 𝑥 𝑤 = 𝑦
2523, 24nfim 1892 . . . . . 6 𝑥((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)
2617, 25nfralw 3305 . . . . 5 𝑥𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)
27 nfv 1910 . . . . 5 𝑤𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)
28 fveqeq2 6906 . . . . . . 7 (𝑤 = 𝑥 → ((𝐹𝑤) = (𝐹𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
29 equequ1 2021 . . . . . . 7 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
3028, 29imbi12d 344 . . . . . 6 (𝑤 = 𝑥 → (((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3130ralbidv 3174 . . . . 5 (𝑤 = 𝑥 → (∀𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦) ↔ ∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3226, 27, 31cbvralw 3300 . . . 4 (∀𝑤𝐴𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3316, 32bitri 275 . . 3 (∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3433anbi2i 622 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣)) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
351, 34bitri 275 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wnfc 2879  wral 3058  wf 6544  1-1wf1 6545  cfv 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fv 6556
This theorem is referenced by:  f1mpt  7271  dom2lem  9013
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