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Theorem f1opr 7461
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 7250 . 2 (𝐹:(𝐴 × 𝐵)–1-1𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤)))
2 fveq2 6888 . . . . . . . . 9 (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹𝑣) = (𝐹‘⟨𝑟, 𝑠⟩))
3 df-ov 7408 . . . . . . . . 9 (𝑟𝐹𝑠) = (𝐹‘⟨𝑟, 𝑠⟩)
42, 3eqtr4di 2790 . . . . . . . 8 (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹𝑣) = (𝑟𝐹𝑠))
54eqeq1d 2734 . . . . . . 7 (𝑣 = ⟨𝑟, 𝑠⟩ → ((𝐹𝑣) = (𝐹𝑤) ↔ (𝑟𝐹𝑠) = (𝐹𝑤)))
6 eqeq1 2736 . . . . . . 7 (𝑣 = ⟨𝑟, 𝑠⟩ → (𝑣 = 𝑤 ↔ ⟨𝑟, 𝑠⟩ = 𝑤))
75, 6imbi12d 344 . . . . . 6 (𝑣 = ⟨𝑟, 𝑠⟩ → (((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤)))
87ralbidv 3177 . . . . 5 (𝑣 = ⟨𝑟, 𝑠⟩ → (∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ∀𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤)))
98ralxp 5839 . . . 4 (∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ∀𝑟𝐴𝑠𝐵𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤))
10 fveq2 6888 . . . . . . . . 9 (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹𝑤) = (𝐹‘⟨𝑡, 𝑢⟩))
11 df-ov 7408 . . . . . . . . 9 (𝑡𝐹𝑢) = (𝐹‘⟨𝑡, 𝑢⟩)
1210, 11eqtr4di 2790 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹𝑤) = (𝑡𝐹𝑢))
1312eqeq2d 2743 . . . . . . 7 (𝑤 = ⟨𝑡, 𝑢⟩ → ((𝑟𝐹𝑠) = (𝐹𝑤) ↔ (𝑟𝐹𝑠) = (𝑡𝐹𝑢)))
14 eqeq2 2744 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑢⟩ → (⟨𝑟, 𝑠⟩ = 𝑤 ↔ ⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩))
15 vex 3478 . . . . . . . . 9 𝑟 ∈ V
16 vex 3478 . . . . . . . . 9 𝑠 ∈ V
1715, 16opth 5475 . . . . . . . 8 (⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩ ↔ (𝑟 = 𝑡𝑠 = 𝑢))
1814, 17bitrdi 286 . . . . . . 7 (𝑤 = ⟨𝑡, 𝑢⟩ → (⟨𝑟, 𝑠⟩ = 𝑤 ↔ (𝑟 = 𝑡𝑠 = 𝑢)))
1913, 18imbi12d 344 . . . . . 6 (𝑤 = ⟨𝑡, 𝑢⟩ → (((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢))))
2019ralxp 5839 . . . . 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 274 . . 3 (∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ∀𝑟𝐴𝑠𝐵𝑡𝐴𝑢𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢)))
2322anbi2i 623 . 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 274 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 205  wa 396   = wceq 1541  wral 3061  cop 4633   × cxp 5673  wf 6536  1-1wf1 6537  cfv 6540  (class class class)co 7405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fv 6548  df-ov 7408
This theorem is referenced by:  fedgmul  32704  aks6d1c2p2  40945
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