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)–1-1→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 7250	. 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 6888	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹‘𝑣) = (𝐹‘⟨𝑟, 𝑠⟩))
3		df-ov 7408	. . . . . . . . 9 ⊢ (𝑟𝐹𝑠) = (𝐹‘⟨𝑟, 𝑠⟩)
4	2, 3	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹‘𝑣) = (𝑟𝐹𝑠))
5	4	eqeq1d 2734	. . . . . . 7 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (𝑟𝐹𝑠) = (𝐹‘𝑤)))
6		eqeq1 2736	. . . . . . 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 3177	. . . . 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 5839	. . . 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 6888	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹‘𝑤) = (𝐹‘⟨𝑡, 𝑢⟩))
11		df-ov 7408	. . . . . . . . 9 ⊢ (𝑡𝐹𝑢) = (𝐹‘⟨𝑡, 𝑢⟩)
12	10, 11	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹‘𝑤) = (𝑡𝐹𝑢))
13	12	eqeq2d 2743	. . . . . . 7 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → ((𝑟𝐹𝑠) = (𝐹‘𝑤) ↔ (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
17	15, 16	opth 5475	. . . . . . . 8 ⊢ (⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩ ↔ (𝑟 = 𝑡 ∧ 𝑠 = 𝑢))
18	14, 17	bitrdi 286	. . . . . . 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 5839	. . . . 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 274	. . 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 274	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 205   ∧ wa 396   = wceq 1541  ∀wral 3061  ⟨cop 4633   × cxp 5673  ⟶wf 6536  –1-1→wf1 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