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Theorem dff13f 6391
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 6390 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣)))
2 dff13f.2 . . . . . . . . 9 𝑦𝐹
3 nfcv 2746 . . . . . . . . 9 𝑦𝑤
42, 3nffv 6091 . . . . . . . 8 𝑦(𝐹𝑤)
5 nfcv 2746 . . . . . . . . 9 𝑦𝑣
62, 5nffv 6091 . . . . . . . 8 𝑦(𝐹𝑣)
74, 6nfeq 2757 . . . . . . 7 𝑦(𝐹𝑤) = (𝐹𝑣)
8 nfv 1828 . . . . . . 7 𝑦 𝑤 = 𝑣
97, 8nfim 1811 . . . . . 6 𝑦((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣)
10 nfv 1828 . . . . . 6 𝑣((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)
11 fveq2 6084 . . . . . . . 8 (𝑣 = 𝑦 → (𝐹𝑣) = (𝐹𝑦))
1211eqeq2d 2615 . . . . . . 7 (𝑣 = 𝑦 → ((𝐹𝑤) = (𝐹𝑣) ↔ (𝐹𝑤) = (𝐹𝑦)))
13 equequ2 1938 . . . . . . 7 (𝑣 = 𝑦 → (𝑤 = 𝑣𝑤 = 𝑦))
1412, 13imbi12d 332 . . . . . 6 (𝑣 = 𝑦 → (((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)))
159, 10, 14cbvral 3138 . . . . 5 (∀𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ∀𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦))
1615ralbii 2958 . . . 4 (∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ∀𝑤𝐴𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦))
17 nfcv 2746 . . . . . 6 𝑥𝐴
18 dff13f.1 . . . . . . . . 9 𝑥𝐹
19 nfcv 2746 . . . . . . . . 9 𝑥𝑤
2018, 19nffv 6091 . . . . . . . 8 𝑥(𝐹𝑤)
21 nfcv 2746 . . . . . . . . 9 𝑥𝑦
2218, 21nffv 6091 . . . . . . . 8 𝑥(𝐹𝑦)
2320, 22nfeq 2757 . . . . . . 7 𝑥(𝐹𝑤) = (𝐹𝑦)
24 nfv 1828 . . . . . . 7 𝑥 𝑤 = 𝑦
2523, 24nfim 1811 . . . . . 6 𝑥((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)
2617, 25nfral 2924 . . . . 5 𝑥𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦)
27 nfv 1828 . . . . 5 𝑤𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)
28 fveq2 6084 . . . . . . . 8 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
2928eqeq1d 2607 . . . . . . 7 (𝑤 = 𝑥 → ((𝐹𝑤) = (𝐹𝑦) ↔ (𝐹𝑥) = (𝐹𝑦)))
30 equequ1 1937 . . . . . . 7 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
3129, 30imbi12d 332 . . . . . 6 (𝑤 = 𝑥 → (((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3231ralbidv 2964 . . . . 5 (𝑤 = 𝑥 → (∀𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦) ↔ ∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3326, 27, 32cbvral 3138 . . . 4 (∀𝑤𝐴𝑦𝐴 ((𝐹𝑤) = (𝐹𝑦) → 𝑤 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3416, 33bitri 262 . . 3 (∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣) ↔ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3534anbi2i 725 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑤𝐴𝑣𝐴 ((𝐹𝑤) = (𝐹𝑣) → 𝑤 = 𝑣)) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
361, 35bitri 262 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wnfc 2733  wral 2891  wf 5782  1-1wf1 5783  cfv 5786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fv 5794
This theorem is referenced by:  f1mpt  6393  dom2lem  7854
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