Theorem f1opr 7489

Description: Condition for an operation to be one-to-one. (Contributed by Jeff Madsen, 17-Jun-2010.)

Assertion

Ref	Expression
f1opr	⊢ (𝐹:(𝐴 × 𝐵)–1-1→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑟 ∈ 𝐴 ∀𝑠 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ∀𝑢 ∈ 𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡 ∧ 𝑠 = 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.

Step	Hyp	Ref	Expression
1		dff13 7275	. 2 ⊢ (𝐹:(𝐴 × 𝐵)–1-1→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤)))
2		fveq2 6906	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹‘𝑣) = (𝐹‘⟨𝑟, 𝑠⟩))
3		df-ov 7434	. . . . . . . . 9 ⊢ (𝑟𝐹𝑠) = (𝐹‘⟨𝑟, 𝑠⟩)
4	2, 3	eqtr4di 2795	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹‘𝑣) = (𝑟𝐹𝑠))
5	4	eqeq1d 2739	. . . . . . 7 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (𝑟𝐹𝑠) = (𝐹‘𝑤)))
6		eqeq1 2741	. . . . . . 7 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → (𝑣 = 𝑤 ↔ ⟨𝑟, 𝑠⟩ = 𝑤))
7	5, 6	imbi12d 344	. . . . . 6 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → (((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤) ↔ ((𝑟𝐹𝑠) = (𝐹‘𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤)))
8	7	ralbidv 3178	. . . . 5 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → (∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤) ↔ ∀𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹‘𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤)))
9	8	ralxp 5852	. . . 4 ⊢ (∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤) ↔ ∀𝑟 ∈ 𝐴 ∀𝑠 ∈ 𝐵 ∀𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹‘𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤))
10		fveq2 6906	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹‘𝑤) = (𝐹‘⟨𝑡, 𝑢⟩))
11		df-ov 7434	. . . . . . . . 9 ⊢ (𝑡𝐹𝑢) = (𝐹‘⟨𝑡, 𝑢⟩)
12	10, 11	eqtr4di 2795	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹‘𝑤) = (𝑡𝐹𝑢))
13	12	eqeq2d 2748	. . . . . . 7 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → ((𝑟𝐹𝑠) = (𝐹‘𝑤) ↔ (𝑟𝐹𝑠) = (𝑡𝐹𝑢)))
14		eqeq2 2749	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (⟨𝑟, 𝑠⟩ = 𝑤 ↔ ⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩))
15		vex 3484	. . . . . . . . 9 ⊢ 𝑟 ∈ V
16		vex 3484	. . . . . . . . 9 ⊢ 𝑠 ∈ V
17	15, 16	opth 5481	. . . . . . . 8 ⊢ (⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩ ↔ (𝑟 = 𝑡 ∧ 𝑠 = 𝑢))
18	14, 17	bitrdi 287	. . . . . . 7 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (⟨𝑟, 𝑠⟩ = 𝑤 ↔ (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))
19	13, 18	imbi12d 344	. . . . . 6 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (((𝑟𝐹𝑠) = (𝐹‘𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢))))
20	19	ralxp 5852	. . . . 5 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹‘𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ∀𝑡 ∈ 𝐴 ∀𝑢 ∈ 𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))
21	20	2ralbii 3128	. . . 4 ⊢ (∀𝑟 ∈ 𝐴 ∀𝑠 ∈ 𝐵 ∀𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹‘𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ∀𝑟 ∈ 𝐴 ∀𝑠 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ∀𝑢 ∈ 𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))
22	9, 21	bitri 275	. . 3 ⊢ (∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤) ↔ ∀𝑟 ∈ 𝐴 ∀𝑠 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ∀𝑢 ∈ 𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))
23	22	anbi2i 623	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑣 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤)) ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑟 ∈ 𝐴 ∀𝑠 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ∀𝑢 ∈ 𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢))))
24	1, 23	bitri 275	1 ⊢ (𝐹:(𝐴 × 𝐵)–1-1→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑟 ∈ 𝐴 ∀𝑠 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ∀𝑢 ∈ 𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∀wral 3061  ⟨cop 4632   × cxp 5683  ⟶wf 6557  –1-1→wf1 6558  ‘cfv 6561  (class class class)co 7431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569  df-ov 7434

This theorem is referenced by:  fedgmul  33682  aks6d1c2p2  42120