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Theorem f1opr 7189
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 6991 . 2 (𝐹:(𝐴 × 𝐵)–1-1𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤)))
2 fveq2 6645 . . . . . . . . 9 (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹𝑣) = (𝐹‘⟨𝑟, 𝑠⟩))
3 df-ov 7138 . . . . . . . . 9 (𝑟𝐹𝑠) = (𝐹‘⟨𝑟, 𝑠⟩)
42, 3eqtr4di 2851 . . . . . . . 8 (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹𝑣) = (𝑟𝐹𝑠))
54eqeq1d 2800 . . . . . . 7 (𝑣 = ⟨𝑟, 𝑠⟩ → ((𝐹𝑣) = (𝐹𝑤) ↔ (𝑟𝐹𝑠) = (𝐹𝑤)))
6 eqeq1 2802 . . . . . . 7 (𝑣 = ⟨𝑟, 𝑠⟩ → (𝑣 = 𝑤 ↔ ⟨𝑟, 𝑠⟩ = 𝑤))
75, 6imbi12d 348 . . . . . 6 (𝑣 = ⟨𝑟, 𝑠⟩ → (((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤)))
87ralbidv 3162 . . . . 5 (𝑣 = ⟨𝑟, 𝑠⟩ → (∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ∀𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤)))
98ralxp 5676 . . . 4 (∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ∀𝑟𝐴𝑠𝐵𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤))
10 fveq2 6645 . . . . . . . . 9 (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹𝑤) = (𝐹‘⟨𝑡, 𝑢⟩))
11 df-ov 7138 . . . . . . . . 9 (𝑡𝐹𝑢) = (𝐹‘⟨𝑡, 𝑢⟩)
1210, 11eqtr4di 2851 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹𝑤) = (𝑡𝐹𝑢))
1312eqeq2d 2809 . . . . . . 7 (𝑤 = ⟨𝑡, 𝑢⟩ → ((𝑟𝐹𝑠) = (𝐹𝑤) ↔ (𝑟𝐹𝑠) = (𝑡𝐹𝑢)))
14 eqeq2 2810 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑢⟩ → (⟨𝑟, 𝑠⟩ = 𝑤 ↔ ⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩))
15 vex 3444 . . . . . . . . 9 𝑟 ∈ V
16 vex 3444 . . . . . . . . 9 𝑠 ∈ V
1715, 16opth 5333 . . . . . . . 8 (⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩ ↔ (𝑟 = 𝑡𝑠 = 𝑢))
1814, 17syl6bb 290 . . . . . . 7 (𝑤 = ⟨𝑡, 𝑢⟩ → (⟨𝑟, 𝑠⟩ = 𝑤 ↔ (𝑟 = 𝑡𝑠 = 𝑢)))
1913, 18imbi12d 348 . . . . . 6 (𝑤 = ⟨𝑡, 𝑢⟩ → (((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢))))
2019ralxp 5676 . . . . 5 (∀𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ∀𝑡𝐴𝑢𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢)))
21202ralbii 3134 . . . 4 (∀𝑟𝐴𝑠𝐵𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ∀𝑟𝐴𝑠𝐵𝑡𝐴𝑢𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢)))
229, 21bitri 278 . . 3 (∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤) ↔ ∀𝑟𝐴𝑠𝐵𝑡𝐴𝑢𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡𝑠 = 𝑢)))
2322anbi2i 625 . 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 278 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 209  wa 399   = wceq 1538  wral 3106  cop 4531   × cxp 5517  wf 6320  1-1wf1 6321  cfv 6324  (class class class)co 7135
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fv 6332  df-ov 7138
This theorem is referenced by:  fedgmul  31115
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