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Theorem tgjustf 26258
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 relopab 5695 . . 3 Rel {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}
2 ancom 463 . . . . . 6 ((𝑥𝐴𝑦𝐴) ↔ (𝑦𝐴𝑥𝐴))
3 eqcom 2828 . . . . . 6 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
42, 3anbi12i 628 . . . . 5 (((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝐹𝑦) = (𝐹𝑥)))
5 simpl 485 . . . . . . . 8 ((𝑢 = 𝑥𝑣 = 𝑦) → 𝑢 = 𝑥)
65fveq2d 6673 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝐹𝑢) = (𝐹𝑥))
7 simpr 487 . . . . . . . 8 ((𝑢 = 𝑥𝑣 = 𝑦) → 𝑣 = 𝑦)
87fveq2d 6673 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝐹𝑣) = (𝐹𝑦))
96, 8eqeq12d 2837 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑦) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑥) = (𝐹𝑦)))
10 eqid 2821 . . . . . 6 {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}
119, 10brab2a 5643 . . . . 5 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)))
12 simpl 485 . . . . . . . 8 ((𝑢 = 𝑦𝑣 = 𝑥) → 𝑢 = 𝑦)
1312fveq2d 6673 . . . . . . 7 ((𝑢 = 𝑦𝑣 = 𝑥) → (𝐹𝑢) = (𝐹𝑦))
14 simpr 487 . . . . . . . 8 ((𝑢 = 𝑦𝑣 = 𝑥) → 𝑣 = 𝑥)
1514fveq2d 6673 . . . . . . 7 ((𝑢 = 𝑦𝑣 = 𝑥) → (𝐹𝑣) = (𝐹𝑥))
1613, 15eqeq12d 2837 . . . . . 6 ((𝑢 = 𝑦𝑣 = 𝑥) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑦) = (𝐹𝑥)))
1716, 10brab2a 5643 . . . . 5 (𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥 ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝐹𝑦) = (𝐹𝑥)))
184, 11, 173bitr4i 305 . . . 4 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥)
1918biimpi 218 . . 3 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥)
20 simplll 773 . . . . 5 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → 𝑥𝐴)
21 simprlr 778 . . . . 5 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → 𝑧𝐴)
22 simplr 767 . . . . . 6 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → (𝐹𝑥) = (𝐹𝑦))
23 simprr 771 . . . . . 6 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → (𝐹𝑦) = (𝐹𝑧))
2422, 23eqtrd 2856 . . . . 5 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → (𝐹𝑥) = (𝐹𝑧))
2520, 21, 24jca31 517 . . . 4 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → ((𝑥𝐴𝑧𝐴) ∧ (𝐹𝑥) = (𝐹𝑧)))
26 simpl 485 . . . . . . . 8 ((𝑢 = 𝑦𝑣 = 𝑧) → 𝑢 = 𝑦)
2726fveq2d 6673 . . . . . . 7 ((𝑢 = 𝑦𝑣 = 𝑧) → (𝐹𝑢) = (𝐹𝑦))
28 simpr 487 . . . . . . . 8 ((𝑢 = 𝑦𝑣 = 𝑧) → 𝑣 = 𝑧)
2928fveq2d 6673 . . . . . . 7 ((𝑢 = 𝑦𝑣 = 𝑧) → (𝐹𝑣) = (𝐹𝑧))
3027, 29eqeq12d 2837 . . . . . 6 ((𝑢 = 𝑦𝑣 = 𝑧) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑦) = (𝐹𝑧)))
3130, 10brab2a 5643 . . . . 5 (𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧 ↔ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧)))
3211, 31anbi12i 628 . . . 4 ((𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧) ↔ (((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))))
33 simpl 485 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑧) → 𝑢 = 𝑥)
3433fveq2d 6673 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑧) → (𝐹𝑢) = (𝐹𝑥))
35 simpr 487 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑧) → 𝑣 = 𝑧)
3635fveq2d 6673 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑧) → (𝐹𝑣) = (𝐹𝑧))
3734, 36eqeq12d 2837 . . . . 5 ((𝑢 = 𝑥𝑣 = 𝑧) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑥) = (𝐹𝑧)))
3837, 10brab2a 5643 . . . 4 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧 ↔ ((𝑥𝐴𝑧𝐴) ∧ (𝐹𝑥) = (𝐹𝑧)))
3925, 32, 383imtr4i 294 . . 3 ((𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧) → 𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧)
40 eqid 2821 . . . . 5 (𝐹𝑥) = (𝐹𝑥)
4140biantru 532 . . . 4 ((𝑥𝐴𝑥𝐴) ↔ ((𝑥𝐴𝑥𝐴) ∧ (𝐹𝑥) = (𝐹𝑥)))
42 pm4.24 566 . . . 4 (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐴))
43 simpl 485 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑥) → 𝑢 = 𝑥)
4443fveq2d 6673 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑥) → (𝐹𝑢) = (𝐹𝑥))
45 simpr 487 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑥) → 𝑣 = 𝑥)
4645fveq2d 6673 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑥) → (𝐹𝑣) = (𝐹𝑥))
4744, 46eqeq12d 2837 . . . . 5 ((𝑢 = 𝑥𝑣 = 𝑥) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑥) = (𝐹𝑥)))
4847, 10brab2a 5643 . . . 4 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥 ↔ ((𝑥𝐴𝑥𝐴) ∧ (𝐹𝑥) = (𝐹𝑥)))
4941, 42, 483bitr4i 305 . . 3 (𝑥𝐴𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥)
501, 19, 39, 49iseri 8315 . 2 {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} Er 𝐴
5111baib 538 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦)))
5251rgen2 3203 . 2 𝑥𝐴𝑦𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))
53 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
54 simprll 777 . . . 4 ((𝐴𝑉 ∧ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))) → 𝑢𝐴)
55 simprlr 778 . . . 4 ((𝐴𝑉 ∧ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))) → 𝑣𝐴)
5653, 53, 54, 55opabex2 7754 . . 3 (𝐴𝑉 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} ∈ V)
57 ereq1 8295 . . . . 5 (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} → (𝑟 Er 𝐴 ↔ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} Er 𝐴))
58 simpl 485 . . . . . . . 8 ((𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} ∧ (𝑥𝐴𝑦𝐴)) → 𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))})
5958breqd 5076 . . . . . . 7 ((𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑟𝑦𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦))
6059bibi1d 346 . . . . . 6 ((𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))))
61602ralbidva 3198 . . . . 5 (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} → (∀𝑥𝐴𝑦𝐴 (𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))))
6257, 61anbi12d 632 . . . 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 3596 . . 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 696 1 (𝐴𝑉 → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wex 1776  wcel 2110  wral 3138  Vcvv 3494   class class class wbr 5065  {copab 5127  cfv 6354   Er wer 8285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fv 6362  df-er 8288
This theorem is referenced by:  tgjustc1  26260
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