Theorem f1opr 33519

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 6512	. 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 6191	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹‘𝑣) = (𝐹‘⟨𝑟, 𝑠⟩))
3		df-ov 6653	. . . . . . . . 9 ⊢ (𝑟𝐹𝑠) = (𝐹‘⟨𝑟, 𝑠⟩)
4	2, 3	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹‘𝑣) = (𝑟𝐹𝑠))
5	4	eqeq1d 2624	. . . . . . 7 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (𝑟𝐹𝑠) = (𝐹‘𝑤)))
6		eqeq1 2626	. . . . . . 7 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → (𝑣 = 𝑤 ↔ ⟨𝑟, 𝑠⟩ = 𝑤))
7	5, 6	imbi12d 334	. . . . . 6 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → (((𝐹‘𝑣) = (𝐹‘𝑤) → 𝑣 = 𝑤) ↔ ((𝑟𝐹𝑠) = (𝐹‘𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤)))
8	7	ralbidv 2986	. . . . 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 5263	. . . 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 6191	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹‘𝑤) = (𝐹‘⟨𝑡, 𝑢⟩))
11		df-ov 6653	. . . . . . . . 9 ⊢ (𝑡𝐹𝑢) = (𝐹‘⟨𝑡, 𝑢⟩)
12	10, 11	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹‘𝑤) = (𝑡𝐹𝑢))
13	12	eqeq2d 2632	. . . . . . 7 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → ((𝑟𝐹𝑠) = (𝐹‘𝑤) ↔ (𝑟𝐹𝑠) = (𝑡𝐹𝑢)))
14		eqeq2 2633	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (⟨𝑟, 𝑠⟩ = 𝑤 ↔ ⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩))
15		vex 3203	. . . . . . . . 9 ⊢ 𝑟 ∈ V
16		vex 3203	. . . . . . . . 9 ⊢ 𝑠 ∈ V
17	15, 16	opth 4945	. . . . . . . 8 ⊢ (⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩ ↔ (𝑟 = 𝑡 ∧ 𝑠 = 𝑢))
18	14, 17	syl6bb 276	. . . . . . 7 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (⟨𝑟, 𝑠⟩ = 𝑤 ↔ (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))
19	13, 18	imbi12d 334	. . . . . 6 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (((𝑟𝐹𝑠) = (𝐹‘𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢))))
20	19	ralxp 5263	. . . . 5 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹‘𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ∀𝑡 ∈ 𝐴 ∀𝑢 ∈ 𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))
21	20	2ralbii 2981	. . . 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 264	. . 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 730	. 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 264	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 196   ∧ wa 384   = wceq 1483  ∀wral 2912  ⟨cop 4183   × cxp 5112  ⟶wf 5884  –1-1→wf1 5885  ‘cfv 5888  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)