Theorem tgjustr 28993

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 8780	. . . . . . 7 ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ 𝑉 → 𝑅 ∈ V))
2	1	impcom 413	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → 𝑅 ∈ V)
3		ecexg 8759	. . . . . 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 3158	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → ∀𝑢 ∈ 𝐴 [𝑢]𝑅 ∈ V)
7		eqid 2763	. . . 4 ⊢ (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)
8	7	fnmpt 6709	. . 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 8810	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
14		eceq1 8795	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → [𝑢]𝑅 = [𝑥]𝑅)
15		ecelqsw 8827	. . . . . . . . 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 7048	. . . . . . 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 8795	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → [𝑢]𝑅 = [𝑦]𝑅)
20		simpr 490	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
21		ecelqsw 8827	. . . . . . . . 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 7048	. . . . . . 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 2779	. . . . 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 3158	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦)))
28	27	ralrimiva 3158	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Er 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦)))
29		mptexg 7258	. . . 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 6660	. . . . 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 6918	. . . . . . . 8 ⊢ ((𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑓‘𝑥) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑥))
34	32	fveq1d 6918	. . . . . . . 8 ⊢ ((𝑓 = (𝑢 ∈ 𝐴 ↦ [𝑢]𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑓‘𝑦) = ((𝑢 ∈ 𝐴 ↦ [𝑢]𝑅)‘𝑦))
35	33, 34	eqeq12d 2779	. . . . . . 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 3228	. . . . 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 3555	. . 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 2147  ∀wral 3080  Vcvv 3454   class class class wbr 5108   ↦ cmpt 5192   Fn wfn 6561  ‘cfv 6566   Er wer 8752  [cec 8753   / cqs 8754

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-er 8755  df-ec 8757  df-qs 8761

This theorem is used by:  tgjustc2  28995