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Theorem tgjustr 27725
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 8727 . . . . . . 7 (𝑅 Er 𝐴 → (𝐴𝑉𝑅 ∈ V))
21impcom 409 . . . . . 6 ((𝐴𝑉𝑅 Er 𝐴) → 𝑅 ∈ V)
3 ecexg 8707 . . . . . 6 (𝑅 ∈ V → [𝑢]𝑅 ∈ V)
42, 3syl 17 . . . . 5 ((𝐴𝑉𝑅 Er 𝐴) → [𝑢]𝑅 ∈ V)
54adantr 482 . . . 4 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑢𝐴) → [𝑢]𝑅 ∈ V)
65ralrimiva 3147 . . 3 ((𝐴𝑉𝑅 Er 𝐴) → ∀𝑢𝐴 [𝑢]𝑅 ∈ V)
7 eqid 2733 . . . 4 (𝑢𝐴 ↦ [𝑢]𝑅) = (𝑢𝐴 ↦ [𝑢]𝑅)
87fnmpt 6691 . . 3 (∀𝑢𝐴 [𝑢]𝑅 ∈ V → (𝑢𝐴 ↦ [𝑢]𝑅) Fn 𝐴)
96, 8syl 17 . 2 ((𝐴𝑉𝑅 Er 𝐴) → (𝑢𝐴 ↦ [𝑢]𝑅) Fn 𝐴)
10 simpllr 775 . . . . . 6 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑅 Er 𝐴)
11 simpr 486 . . . . . . 7 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) → 𝑥𝐴)
1211adantr 482 . . . . . 6 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑥𝐴)
1310, 12erth 8752 . . . . 5 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
14 eceq1 8741 . . . . . . . 8 (𝑢 = 𝑥 → [𝑢]𝑅 = [𝑥]𝑅)
15 ecelqsg 8766 . . . . . . . . 9 ((𝑅 ∈ V ∧ 𝑥𝐴) → [𝑥]𝑅 ∈ (𝐴 / 𝑅))
162, 15sylan 581 . . . . . . . 8 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) → [𝑥]𝑅 ∈ (𝐴 / 𝑅))
177, 14, 11, 16fvmptd3 7022 . . . . . . 7 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) → ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = [𝑥]𝑅)
1817adantr 482 . . . . . 6 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = [𝑥]𝑅)
19 eceq1 8741 . . . . . . . 8 (𝑢 = 𝑦 → [𝑢]𝑅 = [𝑦]𝑅)
20 simpr 486 . . . . . . . 8 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑦𝐴) → 𝑦𝐴)
21 ecelqsg 8766 . . . . . . . . 9 ((𝑅 ∈ V ∧ 𝑦𝐴) → [𝑦]𝑅 ∈ (𝐴 / 𝑅))
222, 21sylan 581 . . . . . . . 8 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑦𝐴) → [𝑦]𝑅 ∈ (𝐴 / 𝑅))
237, 19, 20, 22fvmptd3 7022 . . . . . . 7 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑦𝐴) → ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦) = [𝑦]𝑅)
2423adantlr 714 . . . . . 6 ((((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦) = [𝑦]𝑅)
2518, 24eqeq12d 2749 . . . . 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 3147 . . 3 (((𝐴𝑉𝑅 Er 𝐴) ∧ 𝑥𝐴) → ∀𝑦𝐴 (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦)))
2827ralrimiva 3147 . 2 ((𝐴𝑉𝑅 Er 𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦)))
29 mptexg 7223 . . . 4 (𝐴𝑉 → (𝑢𝐴 ↦ [𝑢]𝑅) ∈ V)
3029adantr 482 . . 3 ((𝐴𝑉𝑅 Er 𝐴) → (𝑢𝐴 ↦ [𝑢]𝑅) ∈ V)
31 fneq1 6641 . . . . 5 (𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) → (𝑓 Fn 𝐴 ↔ (𝑢𝐴 ↦ [𝑢]𝑅) Fn 𝐴))
32 simpl 484 . . . . . . . . 9 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → 𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅))
3332fveq1d 6894 . . . . . . . 8 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → (𝑓𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥))
3432fveq1d 6894 . . . . . . . 8 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → (𝑓𝑦) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦))
3533, 34eqeq12d 2749 . . . . . . 7 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑓𝑥) = (𝑓𝑦) ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦)))
3635bibi2d 343 . . . . . 6 ((𝑓 = (𝑢𝐴 ↦ [𝑢]𝑅) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝑅𝑦 ↔ (𝑓𝑥) = (𝑓𝑦)) ↔ (𝑥𝑅𝑦 ↔ ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢𝐴 ↦ [𝑢]𝑅)‘𝑦))))
37362ralbidva 3217 . . . . 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 3588 . . 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 698 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 397   = wceq 1542  wex 1782  wcel 2107  wral 3062  Vcvv 3475   class class class wbr 5149  cmpt 5232   Fn wfn 6539  cfv 6544   Er wer 8700  [cec 8701   / cqs 8702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-er 8703  df-ec 8705  df-qs 8709
This theorem is referenced by:  tgjustc2  27727
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