Theorem f1opr 7445

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 7229	. 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 6858	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹‘𝑣) = (𝐹‘⟨𝑟, 𝑠⟩))
3		df-ov 7390	. . . . . . . . 9 ⊢ (𝑟𝐹𝑠) = (𝐹‘⟨𝑟, 𝑠⟩)
4	2, 3	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → (𝐹‘𝑣) = (𝑟𝐹𝑠))
5	4	eqeq1d 2731	. . . . . . 7 ⊢ (𝑣 = ⟨𝑟, 𝑠⟩ → ((𝐹‘𝑣) = (𝐹‘𝑤) ↔ (𝑟𝐹𝑠) = (𝐹‘𝑤)))
6		eqeq1 2733	. . . . . . 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 3156	. . . . 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 5805	. . . 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 6858	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹‘𝑤) = (𝐹‘⟨𝑡, 𝑢⟩))
11		df-ov 7390	. . . . . . . . 9 ⊢ (𝑡𝐹𝑢) = (𝐹‘⟨𝑡, 𝑢⟩)
12	10, 11	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝐹‘𝑤) = (𝑡𝐹𝑢))
13	12	eqeq2d 2740	. . . . . . 7 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → ((𝑟𝐹𝑠) = (𝐹‘𝑤) ↔ (𝑟𝐹𝑠) = (𝑡𝐹𝑢)))
14		eqeq2 2741	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (⟨𝑟, 𝑠⟩ = 𝑤 ↔ ⟨𝑟, 𝑠⟩ = ⟨𝑡, 𝑢⟩))
15		vex 3451	. . . . . . . . 9 ⊢ 𝑟 ∈ V
16		vex 3451	. . . . . . . . 9 ⊢ 𝑠 ∈ V
17	15, 16	opth 5436	. . . . . . . 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 5805	. . . . 5 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)((𝑟𝐹𝑠) = (𝐹‘𝑤) → ⟨𝑟, 𝑠⟩ = 𝑤) ↔ ∀𝑡 ∈ 𝐴 ∀𝑢 ∈ 𝐵 ((𝑟𝐹𝑠) = (𝑡𝐹𝑢) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))
21	20	2ralbii 3108	. . . 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 3044  ⟨cop 4595   × cxp 5636  ⟶wf 6507  –1-1→wf1 6508  ‘cfv 6511  (class class class)co 7387

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fv 6519  df-ov 7390

This theorem is referenced by:  fedgmul  33627  aks6d1c2p2  42107