Theorem tgjustr 28482

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	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝑓‘𝑥) = (𝑓‘𝑦))))

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.

Step	Hyp	Ref	Expression
1		erex 8769	. . . . . . 7 ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ 𝑉 → 𝑅 ∈ V))
2	1	impcom 407	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → 𝑅 ∈ V)
3		ecexg 8749	. . . . . 6 ⊢ (𝑅 ∈ V → [𝑢]𝑅 ∈ V)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → [𝑢]𝑅 ∈ V)
5	4	adantr 480	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑢 ∈ 𝐴) → [𝑢]𝑅 ∈ V)
6	5	ralrimiva 3146	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → ∀𝑢 ∈ 𝐴 [𝑢]𝑅 ∈ V)
7		eqid 2737	. . . 4 ⊢ (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)
8	7	fnmpt 6708	. . 3 ⊢ (∀𝑢 ∈ 𝐴 [𝑢]𝑅 ∈ V → (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) Fn 𝐴)
9	6, 8	syl 17	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) Fn 𝐴)
10		simpllr 776	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑅 Er 𝐴)
11		simpr 484	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
12	11	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)
13	10, 12	erth 8796	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
14		eceq1 8784	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → [𝑢]𝑅 = [𝑥]𝑅)
15		ecelqsg 8812	. . . . . . . . 9 ⊢ ((𝑅 ∈ V ∧ 𝑥 ∈ 𝐴) → [𝑥]𝑅 ∈ (𝐴 / 𝑅))
16	2, 15	sylan 580	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) → [𝑥]𝑅 ∈ (𝐴 / 𝑅))
17	7, 14, 11, 16	fvmptd3 7039	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = [𝑥]𝑅)
18	17	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = [𝑥]𝑅)
19		eceq1 8784	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → [𝑢]𝑅 = [𝑦]𝑅)
20		simpr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
21		ecelqsg 8812	. . . . . . . . 9 ⊢ ((𝑅 ∈ V ∧ 𝑦 ∈ 𝐴) → [𝑦]𝑅 ∈ (𝐴 / 𝑅))
22	2, 21	sylan 580	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑦 ∈ 𝐴) → [𝑦]𝑅 ∈ (𝐴 / 𝑅))
23	7, 19, 20, 22	fvmptd3 7039	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦) = [𝑦]𝑅)
24	23	adantlr 715	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦) = [𝑦]𝑅)
25	18, 24	eqeq12d 2753	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦) ↔ [𝑥]𝑅 = [𝑦]𝑅))
26	13, 25	bitr4d 282	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦)))
27	26	ralrimiva 3146	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦)))
28	27	ralrimiva 3146	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦)))
29		mptexg 7241	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∈ V)
30	29	adantr 480	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∈ V)
31		fneq1 6659	. . . . 5 ⊢ (𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) → (𝑓 Fn 𝐴 ↔ (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) Fn 𝐴))
32		simpl 482	. . . . . . . . 9 ⊢ ((𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅))
33	32	fveq1d 6908	. . . . . . . 8 ⊢ ((𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑓‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥))
34	32	fveq1d 6908	. . . . . . . 8 ⊢ ((𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑓‘𝑦) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦))
35	33, 34	eqeq12d 2753	. . . . . . 7 ⊢ ((𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑓‘𝑥) = (𝑓‘𝑦) ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦)))
36	35	bibi2d 342	. . . . . 6 ⊢ ((𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥𝑅𝑦 ↔ (𝑓‘𝑥) = (𝑓‘𝑦)) ↔ (𝑥𝑅𝑦 ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦))))
37	36	2ralbidva 3219	. . . . 5 ⊢ (𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝑓‘𝑥) = (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦))))
38	31, 37	anbi12d 632	. . . 4 ⊢ (𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) → ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝑓‘𝑥) = (𝑓‘𝑦))) ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦)))))
39	38	spcegv 3597	. . 3 ⊢ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∈ V → (((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦))) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝑓‘𝑥) = (𝑓‘𝑦)))))
40	30, 39	syl 17	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → (((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦))) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝑓‘𝑥) = (𝑓‘𝑦)))))
41	9, 28, 40	mp2and 699	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝑓‘𝑥) = (𝑓‘𝑦))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  Vcvv 3480   class class class wbr 5143   ↦ cmpt 5225   Fn wfn 6556  ‘cfv 6561   Er wer 8742  [cec 8743   / cqs 8744

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-er 8745  df-ec 8747  df-qs 8751

This theorem is referenced by:  tgjustc2  28484