Theorem fnrelpredd 32774

Description: A function that preserves a relation also preserves predecessors. (Contributed by BTernaryTau, 16-Jul-2024.)

Hypotheses

Ref	Expression
fnrelpredd.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
fnrelpredd.2	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
fnrelpredd.3	⊢ (𝜑 → 𝐶 ⊆ 𝐴)
fnrelpredd.4	⊢ (𝜑 → 𝐷 ∈ 𝐴)

Assertion

Ref	Expression
fnrelpredd	⊢ (𝜑 → Pred(𝑆, (𝐹 “ 𝐶), (𝐹‘𝐷)) = (𝐹 “ Pred(𝑅, 𝐶, 𝐷)))

Distinct variable groups:   𝑦,𝐴   𝜑,𝑥   𝑥,𝐷,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝐹,𝑦   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑥)   𝐶(𝑦)

Proof of Theorem fnrelpredd

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6730	. . . . 5 ⊢ (𝐹‘𝐷) ∈ V
2	1	dfpred3 6170	. . . 4 ⊢ Pred(𝑆, (𝐹 “ 𝐶), (𝐹‘𝐷)) = {𝑣 ∈ (𝐹 “ 𝐶) ∣ 𝑣𝑆(𝐹‘𝐷)}
3		elrabi 3596	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} → 𝑢 ∈ 𝐶)
4	3	anim1i 618	. . . . . . . . . 10 ⊢ ((𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} ∧ (𝐹‘𝑢) = 𝑣) → (𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢) = 𝑣))
5	4	reximi2 3167	. . . . . . . . 9 ⊢ (∃𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} (𝐹‘𝑢) = 𝑣 → ∃𝑢 ∈ 𝐶 (𝐹‘𝑢) = 𝑣)
6		fnrelpredd.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn 𝐴)
7		fnrelpredd.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
8	6, 7	fvelimabd 6785	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ (𝐹 “ 𝐶) ↔ ∃𝑢 ∈ 𝐶 (𝐹‘𝑢) = 𝑣))
9	5, 8	syl5ibr 249	. . . . . . . 8 ⊢ (𝜑 → (∃𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} (𝐹‘𝑢) = 𝑣 → 𝑣 ∈ (𝐹 “ 𝐶)))
10		fveq2 6717	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝐹‘𝑥) = (𝐹‘𝑢))
11	10	breq1d 5063	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → ((𝐹‘𝑥)𝑆(𝐹‘𝐷) ↔ (𝐹‘𝑢)𝑆(𝐹‘𝐷)))
12	11	elrab 3602	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} ↔ (𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢)𝑆(𝐹‘𝐷)))
13		breq1 5056	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑢) = 𝑣 → ((𝐹‘𝑢)𝑆(𝐹‘𝐷) ↔ 𝑣𝑆(𝐹‘𝐷)))
14	13	biimpac 482	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑢)𝑆(𝐹‘𝐷) ∧ (𝐹‘𝑢) = 𝑣) → 𝑣𝑆(𝐹‘𝐷))
15	14	adantll 714	. . . . . . . . . 10 ⊢ (((𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢)𝑆(𝐹‘𝐷)) ∧ (𝐹‘𝑢) = 𝑣) → 𝑣𝑆(𝐹‘𝐷))
16	12, 15	sylanb 584	. . . . . . . . 9 ⊢ ((𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} ∧ (𝐹‘𝑢) = 𝑣) → 𝑣𝑆(𝐹‘𝐷))
17	16	rexlimiva 3200	. . . . . . . 8 ⊢ (∃𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} (𝐹‘𝑢) = 𝑣 → 𝑣𝑆(𝐹‘𝐷))
18	9, 17	jca2 517	. . . . . . 7 ⊢ (𝜑 → (∃𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} (𝐹‘𝑢) = 𝑣 → (𝑣 ∈ (𝐹 “ 𝐶) ∧ 𝑣𝑆(𝐹‘𝐷))))
19	8	biimpd 232	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ (𝐹 “ 𝐶) → ∃𝑢 ∈ 𝐶 (𝐹‘𝑢) = 𝑣))
20	19	adantrd 495	. . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ (𝐹 “ 𝐶) ∧ 𝑣𝑆(𝐹‘𝐷)) → ∃𝑢 ∈ 𝐶 (𝐹‘𝑢) = 𝑣))
21		simpl 486	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢) = 𝑣) → 𝑢 ∈ 𝐶)
22	21	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑣𝑆(𝐹‘𝐷) → ((𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢) = 𝑣) → 𝑢 ∈ 𝐶))
23	13	biimprcd 253	. . . . . . . . . . . . 13 ⊢ (𝑣𝑆(𝐹‘𝐷) → ((𝐹‘𝑢) = 𝑣 → (𝐹‘𝑢)𝑆(𝐹‘𝐷)))
24	23	adantld 494	. . . . . . . . . . . 12 ⊢ (𝑣𝑆(𝐹‘𝐷) → ((𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢) = 𝑣) → (𝐹‘𝑢)𝑆(𝐹‘𝐷)))
25		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢) = 𝑣) → (𝐹‘𝑢) = 𝑣)
26	25	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑣𝑆(𝐹‘𝐷) → ((𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢) = 𝑣) → (𝐹‘𝑢) = 𝑣))
27	22, 24, 26	3jcad 1131	. . . . . . . . . . 11 ⊢ (𝑣𝑆(𝐹‘𝐷) → ((𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢) = 𝑣) → (𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢)𝑆(𝐹‘𝐷) ∧ (𝐹‘𝑢) = 𝑣)))
28	12	biimpri 231	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢)𝑆(𝐹‘𝐷)) → 𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)})
29	28	anim1i 618	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢)𝑆(𝐹‘𝐷)) ∧ (𝐹‘𝑢) = 𝑣) → (𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} ∧ (𝐹‘𝑢) = 𝑣))
30	29	3impa 1112	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢)𝑆(𝐹‘𝐷) ∧ (𝐹‘𝑢) = 𝑣) → (𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} ∧ (𝐹‘𝑢) = 𝑣))
31	27, 30	syl6 35	. . . . . . . . . 10 ⊢ (𝑣𝑆(𝐹‘𝐷) → ((𝑢 ∈ 𝐶 ∧ (𝐹‘𝑢) = 𝑣) → (𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} ∧ (𝐹‘𝑢) = 𝑣)))
32	31	reximdv2 3190	. . . . . . . . 9 ⊢ (𝑣𝑆(𝐹‘𝐷) → (∃𝑢 ∈ 𝐶 (𝐹‘𝑢) = 𝑣 → ∃𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} (𝐹‘𝑢) = 𝑣))
33	32	adantl 485	. . . . . . . 8 ⊢ ((𝑣 ∈ (𝐹 “ 𝐶) ∧ 𝑣𝑆(𝐹‘𝐷)) → (∃𝑢 ∈ 𝐶 (𝐹‘𝑢) = 𝑣 → ∃𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} (𝐹‘𝑢) = 𝑣))
34	20, 33	sylcom 30	. . . . . . 7 ⊢ (𝜑 → ((𝑣 ∈ (𝐹 “ 𝐶) ∧ 𝑣𝑆(𝐹‘𝐷)) → ∃𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} (𝐹‘𝑢) = 𝑣))
35	18, 34	impbid 215	. . . . . 6 ⊢ (𝜑 → (∃𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} (𝐹‘𝑢) = 𝑣 ↔ (𝑣 ∈ (𝐹 “ 𝐶) ∧ 𝑣𝑆(𝐹‘𝐷))))
36	35	abbidv 2807	. . . . 5 ⊢ (𝜑 → {𝑣 ∣ ∃𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} (𝐹‘𝑢) = 𝑣} = {𝑣 ∣ (𝑣 ∈ (𝐹 “ 𝐶) ∧ 𝑣𝑆(𝐹‘𝐷))})
37		df-rab 3070	. . . . 5 ⊢ {𝑣 ∈ (𝐹 “ 𝐶) ∣ 𝑣𝑆(𝐹‘𝐷)} = {𝑣 ∣ (𝑣 ∈ (𝐹 “ 𝐶) ∧ 𝑣𝑆(𝐹‘𝐷))}
38	36, 37	eqtr4di 2796	. . . 4 ⊢ (𝜑 → {𝑣 ∣ ∃𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} (𝐹‘𝑢) = 𝑣} = {𝑣 ∈ (𝐹 “ 𝐶) ∣ 𝑣𝑆(𝐹‘𝐷)})
39	2, 38	eqtr4id 2797	. . 3 ⊢ (𝜑 → Pred(𝑆, (𝐹 “ 𝐶), (𝐹‘𝐷)) = {𝑣 ∣ ∃𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} (𝐹‘𝑢) = 𝑣})
40		fnfun 6479	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
41	6, 40	syl 17	. . . 4 ⊢ (𝜑 → Fun 𝐹)
42		ssrab2 3993	. . . . . 6 ⊢ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} ⊆ 𝐶
43	42, 7	sstrid 3912	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} ⊆ 𝐴)
44	6	fndmd 6483	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
45	43, 44	sseqtrrd 3942	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} ⊆ dom 𝐹)
46		dfimafn 6775	. . . 4 ⊢ ((Fun 𝐹 ∧ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} ⊆ dom 𝐹) → (𝐹 “ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)}) = {𝑣 ∣ ∃𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} (𝐹‘𝑢) = 𝑣})
47	41, 45, 46	syl2anc 587	. . 3 ⊢ (𝜑 → (𝐹 “ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)}) = {𝑣 ∣ ∃𝑢 ∈ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)} (𝐹‘𝑢) = 𝑣})
48	39, 47	eqtr4d 2780	. 2 ⊢ (𝜑 → Pred(𝑆, (𝐹 “ 𝐶), (𝐹‘𝐷)) = (𝐹 “ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)}))
49		fnrelpredd.4	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
50		dfpred3g 6171	. . . . 5 ⊢ (𝐷 ∈ 𝐴 → Pred(𝑅, 𝐶, 𝐷) = {𝑥 ∈ 𝐶 ∣ 𝑥𝑅𝐷})
51	49, 50	syl 17	. . . 4 ⊢ (𝜑 → Pred(𝑅, 𝐶, 𝐷) = {𝑥 ∈ 𝐶 ∣ 𝑥𝑅𝐷})
52	7	sselda 3901	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐴)
53		fnrelpredd.2	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
54	53	r19.21bi 3130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
55		breq2 5057	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐷 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐷))
56		fveq2 6717	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐷 → (𝐹‘𝑦) = (𝐹‘𝐷))
57	56	breq2d 5065	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐷 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑥)𝑆(𝐹‘𝐷)))
58	55, 57	bibi12d 349	. . . . . . . . . 10 ⊢ (𝑦 = 𝐷 → ((𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)) ↔ (𝑥𝑅𝐷 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝐷))))
59	58	rspcv 3532	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)) → (𝑥𝑅𝐷 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝐷))))
60	49, 59	syl 17	. . . . . . . 8 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)) → (𝑥𝑅𝐷 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝐷))))
61	60	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)) → (𝑥𝑅𝐷 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝐷))))
62	54, 61	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝐷 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝐷)))
63	52, 62	syldan 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥𝑅𝐷 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝐷)))
64	63	rabbidva 3388	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐶 ∣ 𝑥𝑅𝐷} = {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)})
65	51, 64	eqtrd 2777	. . 3 ⊢ (𝜑 → Pred(𝑅, 𝐶, 𝐷) = {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)})
66	65	imaeq2d 5929	. 2 ⊢ (𝜑 → (𝐹 “ Pred(𝑅, 𝐶, 𝐷)) = (𝐹 “ {𝑥 ∈ 𝐶 ∣ (𝐹‘𝑥)𝑆(𝐹‘𝐷)}))
67	48, 66	eqtr4d 2780	1 ⊢ (𝜑 → Pred(𝑆, (𝐹 “ 𝐶), (𝐹‘𝐷)) = (𝐹 “ Pred(𝑅, 𝐶, 𝐷)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  {cab 2714  ∀wral 3061  ∃wrex 3062  {crab 3065   ⊆ wss 3866   class class class wbr 5053  dom cdm 5551   “ cima 5554  Predcpred 6159  Fun wfun 6374   Fn wfn 6375  ‘cfv 6380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388

This theorem is referenced by:  cardpred  32775