Theorem tgjustr 28818

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 8724	. . . . . . 7 ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ 𝑉 → 𝑅 ∈ V))
2	1	impcom 413	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → 𝑅 ∈ V)
3		ecexg 8703	. . . . . 6 ⊢ (𝑅 ∈ V → [𝑢]𝑅 ∈ V)
4	2, 3	syl 18	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → [𝑢]𝑅 ∈ V)
5	4	adantr 486	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑢 ∈ 𝐴) → [𝑢]𝑅 ∈ V)
6	5	ralrimiva 3154	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → ∀𝑢 ∈ 𝐴 [𝑢]𝑅 ∈ V)
7		eqid 2760	. . . 4 ⊢ (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)
8	7	fnmpt 6673	. . 3 ⊢ (∀𝑢 ∈ 𝐴 [𝑢]𝑅 ∈ V → (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) Fn 𝐴)
9	6, 8	syl 18	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) Fn 𝐴)
10		simpllr 788	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑅 Er 𝐴)
11		simpr 490	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
12	11	adantr 486	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)
13	10, 12	erth 8754	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
14		eceq1 8739	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → [𝑢]𝑅 = [𝑥]𝑅)
15		ecelqsw 8771	. . . . . . . . 9 ⊢ ((𝑅 ∈ V ∧ 𝑥 ∈ 𝐴) → [𝑥]𝑅 ∈ (𝐴 / 𝑅))
16	2, 15	sylan 592	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) → [𝑥]𝑅 ∈ (𝐴 / 𝑅))
17	7, 14, 11, 16	fvmptd3 7011	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = [𝑥]𝑅)
18	17	adantr 486	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = [𝑥]𝑅)
19		eceq1 8739	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → [𝑢]𝑅 = [𝑦]𝑅)
20		simpr 490	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
21		ecelqsw 8771	. . . . . . . . 9 ⊢ ((𝑅 ∈ V ∧ 𝑦 ∈ 𝐴) → [𝑦]𝑅 ∈ (𝐴 / 𝑅))
22	2, 21	sylan 592	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑦 ∈ 𝐴) → [𝑦]𝑅 ∈ (𝐴 / 𝑅))
23	7, 19, 20, 22	fvmptd3 7011	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦) = [𝑦]𝑅)
24	23	adantlr 728	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦) = [𝑦]𝑅)
25	18, 24	eqeq12d 2776	. . . . 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 285	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦)))
27	26	ralrimiva 3154	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦)))
28	27	ralrimiva 3154	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦)))
29		mptexg 7221	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∈ V)
30	29	adantr 486	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∈ V)
31		fneq1 6624	. . . . 5 ⊢ (𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) → (𝑓 Fn 𝐴 ↔ (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) Fn 𝐴))
32		simpl 488	. . . . . . . . 9 ⊢ ((𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅))
33	32	fveq1d 6881	. . . . . . . 8 ⊢ ((𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑓‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥))
34	32	fveq1d 6881	. . . . . . . 8 ⊢ ((𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑓‘𝑦) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦))
35	33, 34	eqeq12d 2776	. . . . . . 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 345	. . . . . 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 3224	. . . . 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 644	. . . 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 3551	. . 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 18	. 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 712	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝑓‘𝑥) = (𝑓‘𝑦))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   class class class wbr 5103   ↦ cmpt 5186   Fn wfn 6528  ‘cfv 6533   Er wer 8696  [cec 8697   / cqs 8698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-er 8699  df-ec 8701  df-qs 8705

This theorem is used by:  tgjustc2  28820