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Theorem tgjustr 27992
Description: Given any equivalence relation 𝑅, one can define a function 𝑓 such that all elements of an equivalence classe of 𝑅 have the same image by 𝑓. (Contributed by Thierry Arnoux, 25-Jan-2023.)
Assertion
Ref Expression
tgjustr ((𝐴𝑉𝑅 Er 𝐴) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝑓𝑥) = (𝑓𝑦))))
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝑥,𝑉,𝑦
Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem tgjustr
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 erex 8729 . . . . . . 7 (𝑅 Er 𝐴 → (𝐴𝑉𝑅 ∈ V))
21impcom 406 . . . . . 6 ((𝐴𝑉𝑅 Er 𝐴) → 𝑅 ∈ V)
3 ecexg 8709 . . . . . 6 (𝑅 ∈ V → [𝑢]𝑅 ∈ V)
42, 3syl 17 . . . . 5 ((𝐴𝑉𝑅 Er 𝐴) → [𝑢]𝑅 ∈ V)
54adantr 479 . . . 4 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑢𝐴) → [𝑢]𝑅 ∈ V)
65ralrimiva 3144 . . 3 ((𝐴𝑉𝑅 Er 𝐴) → ∀𝑢𝐴 [𝑢]𝑅 ∈ V)
7 eqid 2730 . . . 4 (𝑢𝐴 ↦ [𝑢]𝑅) = (𝑢𝐴 ↦ [𝑢]𝑅)
87fnmpt 6689 . . 3 (∀𝑢𝐴 [𝑢]𝑅 ∈ V → (𝑢𝐴 ↦ [𝑢]𝑅) Fn 𝐴)
96, 8syl 17 . 2 ((𝐴𝑉𝑅 Er 𝐴) → (𝑢𝐴 ↦ [𝑢]𝑅) Fn 𝐴)
10 simpllr 772 . . . . . 6 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑅 Er 𝐴)
11 simpr 483 . . . . . . 7 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) → 𝑥𝐴)
1211adantr 479 . . . . . 6 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑥𝐴)
1310, 12erth 8754 . . . . 5 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
14 eceq1 8743 . . . . . . . 8 (𝑢 = 𝑥 → [𝑢]𝑅 = [𝑥]𝑅)
15 ecelqsg 8768 . . . . . . . . 9 ((𝑅 ∈ V ∧ 𝑥𝐴) → [𝑥]𝑅 ∈ (𝐴 / 𝑅))
162, 15sylan 578 . . . . . . . 8 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) → [𝑥]𝑅 ∈ (𝐴 / 𝑅))
177, 14, 11, 16fvmptd3 7020 . . . . . . 7 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) → ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = [𝑥]𝑅)
1817adantr 479 . . . . . 6 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = [𝑥]𝑅)
19 eceq1 8743 . . . . . . . 8 (𝑢 = 𝑦 → [𝑢]𝑅 = [𝑦]𝑅)
20 simpr 483 . . . . . . . 8 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑦𝐴) → 𝑦𝐴)
21 ecelqsg 8768 . . . . . . . . 9 ((𝑅 ∈ V ∧ 𝑦𝐴) → [𝑦]𝑅 ∈ (𝐴 / 𝑅))
222, 21sylan 578 . . . . . . . 8 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑦𝐴) → [𝑦]𝑅 ∈ (𝐴 / 𝑅))
237, 19, 20, 22fvmptd3 7020 . . . . . . 7 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑦𝐴) → ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦) = [𝑦]𝑅)
2423adantlr 711 . . . . . 6 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦) = [𝑦]𝑅)
2518, 24eqeq12d 2746 . . . . 5 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → (((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦) ↔ [𝑥]𝑅 = [𝑦]𝑅))
2613, 25bitr4d 281 . . . 4 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦)))
2726ralrimiva 3144 . . 3 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) → ∀𝑦𝐴 (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦)))
2827ralrimiva 3144 . 2 ((𝐴𝑉𝑅 Er 𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦)))
29 mptexg 7224 . . . 4 (𝐴𝑉 → (𝑢𝐴 ↦ [𝑢]𝑅) ∈ V)
3029adantr 479 . . 3 ((𝐴𝑉𝑅 Er 𝐴) → (𝑢𝐴 ↦ [𝑢]𝑅) ∈ V)
31 fneq1 6639 . . . . 5 (𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) → (𝑓 Fn 𝐴 ↔ (𝑢𝐴 ↦ [𝑢]𝑅) Fn 𝐴))
32 simpl 481 . . . . . . . . 9 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → 𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅))
3332fveq1d 6892 . . . . . . . 8 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → (𝑓𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥))
3432fveq1d 6892 . . . . . . . 8 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → (𝑓𝑦) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦))
3533, 34eqeq12d 2746 . . . . . . 7 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑓𝑥) = (𝑓𝑦) ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦)))
3635bibi2d 341 . . . . . 6 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝑅𝑦 ↔ (𝑓𝑥) = (𝑓𝑦)) ↔ (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦))))
37362ralbidva 3214 . . . . 5 (𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝑓𝑥) = (𝑓𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦))))
3831, 37anbi12d 629 . . . 4 (𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) → ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝑓𝑥) = (𝑓𝑦))) ↔ ((𝑢𝐴 ↦ [𝑢]𝑅) Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦)))))
3938spcegv 3586 . . 3 ((𝑢𝐴 ↦ [𝑢]𝑅) ∈ V → (((𝑢𝐴 ↦ [𝑢]𝑅) Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦))) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝑓𝑥) = (𝑓𝑦)))))
4030, 39syl 17 . 2 ((𝐴𝑉𝑅 Er 𝐴) → (((𝑢𝐴 ↦ [𝑢]𝑅) Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦))) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝑓𝑥) = (𝑓𝑦)))))
419, 28, 40mp2and 695 1 ((𝐴𝑉𝑅 Er 𝐴) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝑓𝑥) = (𝑓𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wex 1779  wcel 2104  wral 3059  Vcvv 3472   class class class wbr 5147  cmpt 5230   Fn wfn 6537  cfv 6542   Er wer 8702  [cec 8703   / cqs 8704
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-er 8705  df-ec 8707  df-qs 8711
This theorem is referenced by:  tgjustc2  27994
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