Theorem relpmin 45894

Description: A preimage of a minimal element under a relation-preserving function is minimal. Essentially one half of isomin 7337. (Contributed by Eric Schmidt, 11-Oct-2025.)

Assertion

Ref	Expression
relpmin	⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅ → (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅))

Proof of Theorem relpmin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 4299	. . 3 ⊢ (¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})))
2		relpf 45892	. . . . . . . . . 10 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻:𝐴⟶𝐵)
3	2	ffnd 6702	. . . . . . . . 9 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻 Fn 𝐴)
4		fnfvima 7231	. . . . . . . . . . 11 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶))
5	4	3expia 1139	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑥 ∈ 𝐶 → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
6	5	adantrr 730	. . . . . . . . 9 ⊢ ((𝐻 Fn 𝐴 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ 𝐶 → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
7	3, 6	sylan 592	. . . . . . . 8 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ 𝐶 → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
8	7	adantrd 497	. . . . . . 7 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})) → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
9		ssel 3925	. . . . . . . . . . 11 ⊢ (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
10		vex 3455	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
11	10	eliniseg 6088	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
12	11	ad2antll 742	. . . . . . . . . . . . 13 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
13		relprel 45893	. . . . . . . . . . . . . 14 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 → (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
14		fvex 6890	. . . . . . . . . . . . . . 15 ⊢ (𝐻‘𝐷) ∈ V
15		fvex 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝐻‘𝑥) ∈ V
16	15	eliniseg 6088	. . . . . . . . . . . . . . 15 ⊢ ((𝐻‘𝐷) ∈ V → ((𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
17	14, 16	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}) ↔ (𝐻‘𝑥)𝑆(𝐻‘𝐷))
18	13, 17	imbitrrdi 255	. . . . . . . . . . . . 13 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
19	12, 18	sylbid 243	. . . . . . . . . . . 12 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
20	19	exp32 426	. . . . . . . . . . 11 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝑥 ∈ 𝐴 → (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))))
21	9, 20	syl9r 79	. . . . . . . . . 10 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}))))))
22	21	com34 92	. . . . . . . . 9 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝐶 ⊆ 𝐴 → (𝐷 ∈ 𝐴 → (𝑥 ∈ 𝐶 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}))))))
23	22	imp32 424	. . . . . . . 8 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ 𝐶 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}))))
24	23	impd 416	. . . . . . 7 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
25	8, 24	jcad 522	. . . . . 6 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})) → ((𝐻‘𝑥) ∈ (𝐻 “ 𝐶) ∧ (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}))))
26		elin 3915	. . . . . 6 ⊢ (𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})))
27		elin 3915	. . . . . 6 ⊢ ((𝐻‘𝑥) ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ((𝐻‘𝑥) ∈ (𝐻 “ 𝐶) ∧ (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
28	25, 26, 27	3imtr4g 299	. . . . 5 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) → (𝐻‘𝑥) ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)}))))
29		n0i 4286	. . . . 5 ⊢ ((𝐻‘𝑥) ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) → ¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅)
30	28, 29	syl6 36	. . . 4 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) → ¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))
31	30	exlimdv 1966	. . 3 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (∃𝑥 𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) → ¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))
32	1, 31	biimtrid 245	. 2 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ → ¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))
33	32	con4d 116	1 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅ → (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  {csn 4584   class class class wbr 5103  ^◡ccnv 5650   “ cima 5654   Fn wfn 6526  ‘cfv 6531   RelPres wrelp 45884

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-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-relp 45885

This theorem is used by:  relpfrlem  45895