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Theorem tgjustf 25792
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 5484 . . 3 Rel {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}
2 ancom 454 . . . . . 6 ((𝑥𝐴𝑦𝐴) ↔ (𝑦𝐴𝑥𝐴))
3 eqcom 2832 . . . . . 6 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
42, 3anbi12i 620 . . . . 5 (((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝐹𝑦) = (𝐹𝑥)))
5 simpl 476 . . . . . . . 8 ((𝑢 = 𝑥𝑣 = 𝑦) → 𝑢 = 𝑥)
65fveq2d 6441 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝐹𝑢) = (𝐹𝑥))
7 simpr 479 . . . . . . . 8 ((𝑢 = 𝑥𝑣 = 𝑦) → 𝑣 = 𝑦)
87fveq2d 6441 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝐹𝑣) = (𝐹𝑦))
96, 8eqeq12d 2840 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑦) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑥) = (𝐹𝑦)))
10 eqid 2825 . . . . . 6 {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}
119, 10brab2a 5433 . . . . 5 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)))
12 simpl 476 . . . . . . . 8 ((𝑢 = 𝑦𝑣 = 𝑥) → 𝑢 = 𝑦)
1312fveq2d 6441 . . . . . . 7 ((𝑢 = 𝑦𝑣 = 𝑥) → (𝐹𝑢) = (𝐹𝑦))
14 simpr 479 . . . . . . . 8 ((𝑢 = 𝑦𝑣 = 𝑥) → 𝑣 = 𝑥)
1514fveq2d 6441 . . . . . . 7 ((𝑢 = 𝑦𝑣 = 𝑥) → (𝐹𝑣) = (𝐹𝑥))
1613, 15eqeq12d 2840 . . . . . 6 ((𝑢 = 𝑦𝑣 = 𝑥) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑦) = (𝐹𝑥)))
1716, 10brab2a 5433 . . . . 5 (𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥 ↔ ((𝑦𝐴𝑥𝐴) ∧ (𝐹𝑦) = (𝐹𝑥)))
184, 11, 173bitr4i 295 . . . 4 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥)
1918biimpi 208 . . 3 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥)
20 simplll 791 . . . . . 6 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → 𝑥𝐴)
21 simprlr 798 . . . . . 6 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → 𝑧𝐴)
2220, 21jca 507 . . . . 5 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → (𝑥𝐴𝑧𝐴))
23 simplr 785 . . . . . 6 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → (𝐹𝑥) = (𝐹𝑦))
24 simprr 789 . . . . . 6 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → (𝐹𝑦) = (𝐹𝑧))
2523, 24eqtrd 2861 . . . . 5 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → (𝐹𝑥) = (𝐹𝑧))
2622, 25jca 507 . . . 4 ((((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))) → ((𝑥𝐴𝑧𝐴) ∧ (𝐹𝑥) = (𝐹𝑧)))
27 simpl 476 . . . . . . . 8 ((𝑢 = 𝑦𝑣 = 𝑧) → 𝑢 = 𝑦)
2827fveq2d 6441 . . . . . . 7 ((𝑢 = 𝑦𝑣 = 𝑧) → (𝐹𝑢) = (𝐹𝑦))
29 simpr 479 . . . . . . . 8 ((𝑢 = 𝑦𝑣 = 𝑧) → 𝑣 = 𝑧)
3029fveq2d 6441 . . . . . . 7 ((𝑢 = 𝑦𝑣 = 𝑧) → (𝐹𝑣) = (𝐹𝑧))
3128, 30eqeq12d 2840 . . . . . 6 ((𝑢 = 𝑦𝑣 = 𝑧) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑦) = (𝐹𝑧)))
3231, 10brab2a 5433 . . . . 5 (𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧 ↔ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧)))
3311, 32anbi12i 620 . . . 4 ((𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧) ↔ (((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) ∧ ((𝑦𝐴𝑧𝐴) ∧ (𝐹𝑦) = (𝐹𝑧))))
34 simpl 476 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑧) → 𝑢 = 𝑥)
3534fveq2d 6441 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑧) → (𝐹𝑢) = (𝐹𝑥))
36 simpr 479 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑧) → 𝑣 = 𝑧)
3736fveq2d 6441 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑧) → (𝐹𝑣) = (𝐹𝑧))
3835, 37eqeq12d 2840 . . . . 5 ((𝑢 = 𝑥𝑣 = 𝑧) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑥) = (𝐹𝑧)))
3938, 10brab2a 5433 . . . 4 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧 ↔ ((𝑥𝐴𝑧𝐴) ∧ (𝐹𝑥) = (𝐹𝑧)))
4026, 33, 393imtr4i 284 . . 3 ((𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦𝑦{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧) → 𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑧)
41 pm4.24 559 . . . . 5 (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐴))
42 eqid 2825 . . . . . 6 (𝐹𝑥) = (𝐹𝑥)
4342biantru 525 . . . . 5 ((𝑥𝐴𝑥𝐴) ↔ ((𝑥𝐴𝑥𝐴) ∧ (𝐹𝑥) = (𝐹𝑥)))
4441, 43bitri 267 . . . 4 (𝑥𝐴 ↔ ((𝑥𝐴𝑥𝐴) ∧ (𝐹𝑥) = (𝐹𝑥)))
45 simpl 476 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑥) → 𝑢 = 𝑥)
4645fveq2d 6441 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑥) → (𝐹𝑢) = (𝐹𝑥))
47 simpr 479 . . . . . . 7 ((𝑢 = 𝑥𝑣 = 𝑥) → 𝑣 = 𝑥)
4847fveq2d 6441 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑥) → (𝐹𝑣) = (𝐹𝑥))
4946, 48eqeq12d 2840 . . . . 5 ((𝑢 = 𝑥𝑣 = 𝑥) → ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑥) = (𝐹𝑥)))
5049, 10brab2a 5433 . . . 4 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥 ↔ ((𝑥𝐴𝑥𝐴) ∧ (𝐹𝑥) = (𝐹𝑥)))
5144, 50bitr4i 270 . . 3 (𝑥𝐴𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑥)
521, 19, 40, 51iseri 8041 . 2 {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} Er 𝐴
5311baib 531 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦)))
5453rgen2 3184 . 2 𝑥𝐴𝑦𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))
55 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
56 simprll 797 . . . 4 ((𝐴𝑉 ∧ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))) → 𝑢𝐴)
57 simprlr 798 . . . 4 ((𝐴𝑉 ∧ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))) → 𝑣𝐴)
5855, 55, 56, 57opabex2 7494 . . 3 (𝐴𝑉 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} ∈ V)
59 ereq1 8021 . . . . 5 (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} → (𝑟 Er 𝐴 ↔ {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} Er 𝐴))
60 simpl 476 . . . . . . . 8 ((𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} ∧ (𝑥𝐴𝑦𝐴)) → 𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))})
6160breqd 4886 . . . . . . 7 ((𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑟𝑦𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦))
6261bibi1d 335 . . . . . 6 ((𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))))
63622ralbidva 3197 . . . . 5 (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))} → (∀𝑥𝐴𝑦𝐴 (𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥{⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐴) ∧ (𝐹𝑢) = (𝐹𝑣))}𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))))
6459, 63anbi12d 624 . . . 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)))))
6564spcegv 3511 . . 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)))))
6658, 65syl 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)))))
6752, 54, 66mp2ani 689 1 (𝐴𝑉 → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wex 1878  wcel 2164  wral 3117  Vcvv 3414   class class class wbr 4875  {copab 4937  cfv 6127   Er wer 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fv 6135  df-er 8014
This theorem is referenced by:  tgjustc1  25794
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