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Theorem tgjustf 26596
Description: Given any function 𝐹, equality of the image by 𝐹 is an equivalence relation. (Contributed by Thierry Arnoux, 25-Jan-2023.)
Assertion
Ref Expression
tgjustf (𝐴𝑉 → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))))
Distinct variable groups:   𝐴,𝑟,𝑥,𝑦   𝐹,𝑟,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑟)

Proof of Theorem tgjustf
Dummy variables 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopabv 5709 . . 3 Rel {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}
2 ancom 464 . . . . . 6 ((𝑥𝐴𝑦𝐴) ↔ (𝑦𝐴𝑥𝐴))
3 eqcom 2746 . . . . . 6 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
42, 3anbi12i 630 . . . . 5 (((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝐹𝑦) = (𝐹𝑥)))
5 simpl 486 . . . . . . . 8 ((𝑢 = 𝑥𝑣 = 𝑦) → 𝑢 = 𝑥)
65fveq2d 6743 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝐹𝑢) = (𝐹𝑥))
7 simpr 488 . . . . . . . 8 ((𝑢 = 𝑥𝑣 = 𝑦) → 𝑣 = 𝑦)
87fveq2d 6743 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝐹𝑣) = (𝐹𝑦))
96, 8eqeq12d 2755 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑦) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑥) = (𝐹𝑦)))
10 eqid 2739 . . . . . 6 {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}
119, 10brab2a 5659 . . . . 5 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)))
12 simpl 486 . . . . . . . 8 ((𝑢 = 𝑦𝑣 = 𝑥) → 𝑢 = 𝑦)
1312fveq2d 6743 . . . . . . 7 ((𝑢 = 𝑦𝑣 = 𝑥) → (𝐹𝑢) = (𝐹𝑦))
14 simpr 488 . . . . . . . 8 ((𝑢 = 𝑦𝑣 = 𝑥) → 𝑣 = 𝑥)
1514fveq2d 6743 . . . . . . 7 ((𝑢 = 𝑦𝑣 = 𝑥) → (𝐹𝑣) = (𝐹𝑥))
1613, 15eqeq12d 2755 . . . . . 6 ((𝑢 = 𝑦𝑣 = 𝑥) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑦) = (𝐹𝑥)))
1716, 10brab2a 5659 . . . . 5 (𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥 ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝐹𝑦) = (𝐹𝑥)))
184, 11, 173bitr4i 306 . . . 4 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥)
1918biimpi 219 . . 3 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥)
20 simplll 775 . . . . 5 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → 𝑥𝐴)
21 simprlr 780 . . . . 5 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → 𝑧𝐴)
22 simplr 769 . . . . . 6 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → (𝐹𝑥) = (𝐹𝑦))
23 simprr 773 . . . . . 6 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → (𝐹𝑦) = (𝐹𝑧))
2422, 23eqtrd 2779 . . . . 5 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → (𝐹𝑥) = (𝐹𝑧))
2520, 21, 24jca31 518 . . . 4 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → ((𝑥𝐴𝑧𝐴) ∧ (𝐹𝑥) = (𝐹𝑧)))
26 simpl 486 . . . . . . . 8 ((𝑢 = 𝑦𝑣 = 𝑧) → 𝑢 = 𝑦)
2726fveq2d 6743 . . . . . . 7 ((𝑢 = 𝑦𝑣 = 𝑧) → (𝐹𝑢) = (𝐹𝑦))
28 simpr 488 . . . . . . . 8 ((𝑢 = 𝑦𝑣 = 𝑧) → 𝑣 = 𝑧)
2928fveq2d 6743 . . . . . . 7 ((𝑢 = 𝑦𝑣 = 𝑧) → (𝐹𝑣) = (𝐹𝑧))
3027, 29eqeq12d 2755 . . . . . 6 ((𝑢 = 𝑦𝑣 = 𝑧) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑦) = (𝐹𝑧)))
3130, 10brab2a 5659 . . . . 5 (𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧 ↔ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧)))
3211, 31anbi12i 630 . . . 4 ((𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧) ↔ (((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))))
33 simpl 486 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑧) → 𝑢 = 𝑥)
3433fveq2d 6743 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑧) → (𝐹𝑢) = (𝐹𝑥))
35 simpr 488 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑧) → 𝑣 = 𝑧)
3635fveq2d 6743 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑧) → (𝐹𝑣) = (𝐹𝑧))
3734, 36eqeq12d 2755 . . . . 5 ((𝑢 = 𝑥𝑣 = 𝑧) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑥) = (𝐹𝑧)))
3837, 10brab2a 5659 . . . 4 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧 ↔ ((𝑥𝐴𝑧𝐴) ∧ (𝐹𝑥) = (𝐹𝑧)))
3925, 32, 383imtr4i 295 . . 3 ((𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧) → 𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧)
40 eqid 2739 . . . . 5 (𝐹𝑥) = (𝐹𝑥)
4140biantru 533 . . . 4 ((𝑥𝐴𝑥𝐴) ↔ ((𝑥𝐴𝑥𝐴) ∧ (𝐹𝑥) = (𝐹𝑥)))
42 pm4.24 567 . . . 4 (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐴))
43 simpl 486 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑥) → 𝑢 = 𝑥)
4443fveq2d 6743 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑥) → (𝐹𝑢) = (𝐹𝑥))
45 simpr 488 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑥) → 𝑣 = 𝑥)
4645fveq2d 6743 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑥) → (𝐹𝑣) = (𝐹𝑥))
4744, 46eqeq12d 2755 . . . . 5 ((𝑢 = 𝑥𝑣 = 𝑥) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑥) = (𝐹𝑥)))
4847, 10brab2a 5659 . . . 4 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥 ↔ ((𝑥𝐴𝑥𝐴) ∧ (𝐹𝑥) = (𝐹𝑥)))
4941, 42, 483bitr4i 306 . . 3 (𝑥𝐴𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥)
501, 19, 39, 49iseri 8442 . 2 {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} Er 𝐴
5111baib 539 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦)))
5251rgen2 3127 . 2 𝑥𝐴𝑦𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))
53 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
54 simprll 779 . . . 4 ((𝐴𝑉 ∧ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))) → 𝑢𝐴)
55 simprlr 780 . . . 4 ((𝐴𝑉 ∧ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))) → 𝑣𝐴)
5653, 53, 54, 55opabex2 7849 . . 3 (𝐴𝑉 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} ∈ V)
57 ereq1 8422 . . . . 5 (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} → (𝑟 Er 𝐴 ↔ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} Er 𝐴))
58 simpl 486 . . . . . . . 8 ((𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} ∧ (𝑥𝐴𝑦𝐴)) → 𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))})
5958breqd 5081 . . . . . . 7 ((𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑟𝑦𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦))
6059bibi1d 347 . . . . . 6 ((𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))))
61602ralbidva 3122 . . . . 5 (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} → (∀𝑥𝐴𝑦𝐴 (𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))))
6257, 61anbi12d 634 . . . 4 (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} → ((𝑟 Er 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))) ↔ ({⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} Er 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦)))))
6362spcegv 3527 . . 3 ({⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} ∈ V → (({⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} Er 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))) → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦)))))
6456, 63syl 17 . 2 (𝐴𝑉 → (({⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} Er 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))) → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦)))))
6550, 52, 64mp2ani 698 1 (𝐴𝑉 → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2112  wral 3064  Vcvv 3423   class class class wbr 5070  {copab 5132  cfv 6401   Er wer 8412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5209  ax-nul 5216  ax-pow 5275  ax-pr 5339  ax-un 7545
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5575  df-rel 5576  df-cnv 5577  df-co 5578  df-dm 5579  df-iota 6359  df-fv 6409  df-er 8415
This theorem is referenced by:  tgjustc1  26598
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