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Theorem tgjustr 28496
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 8767 . . . . . . 7 (𝑅 Er 𝐴 → (𝐴𝑉𝑅 ∈ V))
21impcom 407 . . . . . 6 ((𝐴𝑉𝑅 Er 𝐴) → 𝑅 ∈ V)
3 ecexg 8747 . . . . . 6 (𝑅 ∈ V → [𝑢]𝑅 ∈ V)
42, 3syl 17 . . . . 5 ((𝐴𝑉𝑅 Er 𝐴) → [𝑢]𝑅 ∈ V)
54adantr 480 . . . 4 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑢𝐴) → [𝑢]𝑅 ∈ V)
65ralrimiva 3143 . . 3 ((𝐴𝑉𝑅 Er 𝐴) → ∀𝑢𝐴 [𝑢]𝑅 ∈ V)
7 eqid 2734 . . . 4 (𝑢𝐴 ↦ [𝑢]𝑅) = (𝑢𝐴 ↦ [𝑢]𝑅)
87fnmpt 6708 . . 3 (∀𝑢𝐴 [𝑢]𝑅 ∈ V → (𝑢𝐴 ↦ [𝑢]𝑅) Fn 𝐴)
96, 8syl 17 . 2 ((𝐴𝑉𝑅 Er 𝐴) → (𝑢𝐴 ↦ [𝑢]𝑅) Fn 𝐴)
10 simpllr 776 . . . . . 6 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑅 Er 𝐴)
11 simpr 484 . . . . . . 7 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) → 𝑥𝐴)
1211adantr 480 . . . . . 6 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑥𝐴)
1310, 12erth 8794 . . . . 5 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
14 eceq1 8782 . . . . . . . 8 (𝑢 = 𝑥 → [𝑢]𝑅 = [𝑥]𝑅)
15 ecelqsg 8810 . . . . . . . . 9 ((𝑅 ∈ V ∧ 𝑥𝐴) → [𝑥]𝑅 ∈ (𝐴 / 𝑅))
162, 15sylan 580 . . . . . . . 8 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) → [𝑥]𝑅 ∈ (𝐴 / 𝑅))
177, 14, 11, 16fvmptd3 7038 . . . . . . 7 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) → ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = [𝑥]𝑅)
1817adantr 480 . . . . . 6 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = [𝑥]𝑅)
19 eceq1 8782 . . . . . . . 8 (𝑢 = 𝑦 → [𝑢]𝑅 = [𝑦]𝑅)
20 simpr 484 . . . . . . . 8 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑦𝐴) → 𝑦𝐴)
21 ecelqsg 8810 . . . . . . . . 9 ((𝑅 ∈ V ∧ 𝑦𝐴) → [𝑦]𝑅 ∈ (𝐴 / 𝑅))
222, 21sylan 580 . . . . . . . 8 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑦𝐴) → [𝑦]𝑅 ∈ (𝐴 / 𝑅))
237, 19, 20, 22fvmptd3 7038 . . . . . . 7 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑦𝐴) → ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦) = [𝑦]𝑅)
2423adantlr 715 . . . . . 6 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦) = [𝑦]𝑅)
2518, 24eqeq12d 2750 . . . . 5 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → (((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦) ↔ [𝑥]𝑅 = [𝑦]𝑅))
2613, 25bitr4d 282 . . . 4 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦)))
2726ralrimiva 3143 . . 3 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) → ∀𝑦𝐴 (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦)))
2827ralrimiva 3143 . 2 ((𝐴𝑉𝑅 Er 𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦)))
29 mptexg 7240 . . . 4 (𝐴𝑉 → (𝑢𝐴 ↦ [𝑢]𝑅) ∈ V)
3029adantr 480 . . 3 ((𝐴𝑉𝑅 Er 𝐴) → (𝑢𝐴 ↦ [𝑢]𝑅) ∈ V)
31 fneq1 6659 . . . . 5 (𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) → (𝑓 Fn 𝐴 ↔ (𝑢𝐴 ↦ [𝑢]𝑅) Fn 𝐴))
32 simpl 482 . . . . . . . . 9 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → 𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅))
3332fveq1d 6908 . . . . . . . 8 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → (𝑓𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥))
3432fveq1d 6908 . . . . . . . 8 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → (𝑓𝑦) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦))
3533, 34eqeq12d 2750 . . . . . . 7 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑓𝑥) = (𝑓𝑦) ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦)))
3635bibi2d 342 . . . . . 6 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝑅𝑦 ↔ (𝑓𝑥) = (𝑓𝑦)) ↔ (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦))))
37362ralbidva 3216 . . . . 5 (𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝑓𝑥) = (𝑓𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦))))
3831, 37anbi12d 632 . . . 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 3596 . . 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 699 1 ((𝐴𝑉𝑅 Er 𝐴) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝑓𝑥) = (𝑓𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  wral 3058  Vcvv 3477   class class class wbr 5147  cmpt 5230   Fn wfn 6557  cfv 6562   Er wer 8740  [cec 8741   / cqs 8742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-er 8743  df-ec 8745  df-qs 8749
This theorem is referenced by:  tgjustc2  28498
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