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Theorem f1opr 33151
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 6466 . 2 (𝐹:(𝐴 × 𝐵)–1-1𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤)))
2 fveq2 6148 . . . . . . . . 9 (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹𝑣) = (𝐹‘⟨𝑟, 𝑠⟩))
3 df-ov 6607 . . . . . . . . 9 (𝑟𝐹𝑠) = (𝐹‘⟨𝑟, 𝑠⟩)
42, 3syl6eqr 2673 . . . . . . . 8 (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹𝑣) = (𝑟𝐹𝑠))
54eqeq1d 2623 . . . . . . 7 (𝑣 = ⟨𝑟, 𝑠⟩ → ((𝐹𝑣) = (𝐹𝑤) ↔ (𝑟𝐹𝑠) = (𝐹𝑤)))
6 eqeq1 2625 . . . . . . 7 (𝑣 = ⟨𝑟, 𝑠⟩ → (𝑣 = 𝑤 ↔ ⟨𝑟, 𝑠⟩ = 𝑤))
75, 6imbi12d 334 . . . . . 6 (𝑣 = ⟨𝑟, 𝑠⟩ → (((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤)))
87ralbidv 2980 . . . . 5 (𝑣 = ⟨𝑟, 𝑠⟩ → (∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ∀𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤)))
98ralxp 5223 . . . 4 (∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ∀𝑟𝐴𝑠𝐵𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤))
10 fveq2 6148 . . . . . . . . 9 (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹𝑤) = (𝐹‘⟨𝑡, 𝑢⟩))
11 df-ov 6607 . . . . . . . . 9 (𝑡𝐹𝑢) = (𝐹‘⟨𝑡, 𝑢⟩)
1210, 11syl6eqr 2673 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹𝑤) = (𝑡𝐹𝑢))
1312eqeq2d 2631 . . . . . . 7 (𝑤 = ⟨𝑡, 𝑢⟩ → ((𝑟𝐹𝑠) = (𝐹𝑤) ↔ (𝑟𝐹𝑠) = (𝑡𝐹𝑢)))
14 eqeq2 2632 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑢⟩ → (⟨𝑟, 𝑠⟩ = 𝑤 ↔ ⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩))
15 vex 3189 . . . . . . . . 9 𝑟 ∈ V
16 vex 3189 . . . . . . . . 9 𝑠 ∈ V
1715, 16opth 4905 . . . . . . . 8 (⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩ ↔ (𝑟 = 𝑡𝑠 = 𝑢))
1814, 17syl6bb 276 . . . . . . 7 (𝑤 = ⟨𝑡, 𝑢⟩ → (⟨𝑟, 𝑠⟩ = 𝑤 ↔ (𝑟 = 𝑡𝑠 = 𝑢)))
1913, 18imbi12d 334 . . . . . 6 (𝑤 = ⟨𝑡, 𝑢⟩ → (((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢))))
2019ralxp 5223 . . . . 5 (∀𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ∀𝑡𝐴𝑢𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢)))
21202ralbii 2975 . . . 4 (∀𝑟𝐴𝑠𝐵𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ∀𝑟𝐴𝑠𝐵𝑡𝐴𝑢𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢)))
229, 21bitri 264 . . 3 (∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ∀𝑟𝐴𝑠𝐵𝑡𝐴𝑢𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢)))
2322anbi2i 729 . 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 264 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 196  wa 384   = wceq 1480  wral 2907  cop 4154   × cxp 5072  wf 5843  1-1wf1 5844  cfv 5847  (class class class)co 6604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fv 5855  df-ov 6607
This theorem is referenced by: (None)
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