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Theorem f1opr 33874
Description: Condition for an operation to be one-to-one. (Contributed by Jeff Madsen, 17-Jun-2010.)
Assertion
Ref Expression
f1opr (𝐹:(𝐴 × 𝐵)–1-1𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑟𝐴𝑠𝐵𝑡𝐴𝑢𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢))))
Distinct variable groups:   𝐴,𝑟,𝑠,𝑡,𝑢   𝐵,𝑟,𝑠,𝑡,𝑢   𝐹,𝑟,𝑠,𝑡,𝑢
Allowed substitution hints:   𝐶(𝑢,𝑡,𝑠,𝑟)

Proof of Theorem f1opr
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dff13 6704 . 2 (𝐹:(𝐴 × 𝐵)–1-1𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤)))
2 fveq2 6375 . . . . . . . . 9 (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹𝑣) = (𝐹‘⟨𝑟, 𝑠⟩))
3 df-ov 6845 . . . . . . . . 9 (𝑟𝐹𝑠) = (𝐹‘⟨𝑟, 𝑠⟩)
42, 3syl6eqr 2817 . . . . . . . 8 (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹𝑣) = (𝑟𝐹𝑠))
54eqeq1d 2767 . . . . . . 7 (𝑣 = ⟨𝑟, 𝑠⟩ → ((𝐹𝑣) = (𝐹𝑤) ↔ (𝑟𝐹𝑠) = (𝐹𝑤)))
6 eqeq1 2769 . . . . . . 7 (𝑣 = ⟨𝑟, 𝑠⟩ → (𝑣 = 𝑤 ↔ ⟨𝑟, 𝑠⟩ = 𝑤))
75, 6imbi12d 335 . . . . . 6 (𝑣 = ⟨𝑟, 𝑠⟩ → (((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤)))
87ralbidv 3133 . . . . 5 (𝑣 = ⟨𝑟, 𝑠⟩ → (∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ∀𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤)))
98ralxp 5432 . . . 4 (∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ∀𝑟𝐴𝑠𝐵𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤))
10 fveq2 6375 . . . . . . . . 9 (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹𝑤) = (𝐹‘⟨𝑡, 𝑢⟩))
11 df-ov 6845 . . . . . . . . 9 (𝑡𝐹𝑢) = (𝐹‘⟨𝑡, 𝑢⟩)
1210, 11syl6eqr 2817 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹𝑤) = (𝑡𝐹𝑢))
1312eqeq2d 2775 . . . . . . 7 (𝑤 = ⟨𝑡, 𝑢⟩ → ((𝑟𝐹𝑠) = (𝐹𝑤) ↔ (𝑟𝐹𝑠) = (𝑡𝐹𝑢)))
14 eqeq2 2776 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑢⟩ → (⟨𝑟, 𝑠⟩ = 𝑤 ↔ ⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩))
15 vex 3353 . . . . . . . . 9 𝑟 ∈ V
16 vex 3353 . . . . . . . . 9 𝑠 ∈ V
1715, 16opth 5100 . . . . . . . 8 (⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩ ↔ (𝑟 = 𝑡𝑠 = 𝑢))
1814, 17syl6bb 278 . . . . . . 7 (𝑤 = ⟨𝑡, 𝑢⟩ → (⟨𝑟, 𝑠⟩ = 𝑤 ↔ (𝑟 = 𝑡𝑠 = 𝑢)))
1913, 18imbi12d 335 . . . . . 6 (𝑤 = ⟨𝑡, 𝑢⟩ → (((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢))))
2019ralxp 5432 . . . . 5 (∀𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ∀𝑡𝐴𝑢𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢)))
21202ralbii 3128 . . . 4 (∀𝑟𝐴𝑠𝐵𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ∀𝑟𝐴𝑠𝐵𝑡𝐴𝑢𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢)))
229, 21bitri 266 . . 3 (∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ∀𝑟𝐴𝑠𝐵𝑡𝐴𝑢𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢)))
2322anbi2i 616 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤)) ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑟𝐴𝑠𝐵𝑡𝐴𝑢𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢))))
241, 23bitri 266 1 (𝐹:(𝐴 × 𝐵)–1-1𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑟𝐴𝑠𝐵𝑡𝐴𝑢𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wral 3055  cop 4340   × cxp 5275  wf 6064  1-1wf1 6065  cfv 6068  (class class class)co 6842
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 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
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 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fv 6076  df-ov 6845
This theorem is referenced by: (None)
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