Theorem relpmin 44914

Description: A preimage of a minimal element under a relation-preserving function is minimal. Essentially one half of isomin 7319. (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 4323	. . 3 ⊢ (¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})))
2		relpf 44912	. . . . . . . . . 10 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻:𝐴⟶𝐵)
3	2	ffnd 6696	. . . . . . . . 9 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻 Fn 𝐴)
4		fnfvima 7214	. . . . . . . . . . 11 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶))
5	4	3expia 1121	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑥 ∈ 𝐶 → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
6	5	adantrr 717	. . . . . . . . 9 ⊢ ((𝐻 Fn 𝐴 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ 𝐶 → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
7	3, 6	sylan 580	. . . . . . . 8 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ 𝐶 → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
8	7	adantrd 491	. . . . . . 7 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})) → (𝐻‘𝑥) ∈ (𝐻 “ 𝐶)))
9		ssel 3948	. . . . . . . . . . 11 ⊢ (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
10		vex 3459	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
11	10	eliniseg 6073	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
12	11	ad2antll 729	. . . . . . . . . . . . 13 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (◡𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
13		relprel 44913	. . . . . . . . . . . . . 14 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 → (𝐻‘𝑥)𝑆(𝐻‘𝐷)))
14		fvex 6878	. . . . . . . . . . . . . . 15 ⊢ (𝐻‘𝐷) ∈ V
15		fvex 6878	. . . . . . . . . . . . . . . 16 ⊢ (𝐻‘𝑥) ∈ V
16	15	eliniseg 6073	. . . . . . . . . . . . . . 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 252	. . . . . . . . . . . . 13 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥𝑅𝐷 → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
19	12, 18	sylbid 240	. . . . . . . . . . . 12 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
20	19	exp32 420	. . . . . . . . . . 11 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝑥 ∈ 𝐴 → (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))))
21	9, 20	syl9r 78	. . . . . . . . . 10 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝐶 ⊆ 𝐴 → (𝑥 ∈ 𝐶 → (𝐷 ∈ 𝐴 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}))))))
22	21	com34 91	. . . . . . . . 9 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝐶 ⊆ 𝐴 → (𝐷 ∈ 𝐴 → (𝑥 ∈ 𝐶 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}))))))
23	22	imp32 418	. . . . . . . 8 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ 𝐶 → (𝑥 ∈ (◡𝑅 “ {𝐷}) → (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)}))))
24	23	impd 410	. . . . . . 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 512	. . . . . 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 3938	. . . . . 6 ⊢ (𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ∈ (◡𝑅 “ {𝐷})))
27		elin 3938	. . . . . 6 ⊢ ((𝐻‘𝑥) ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) ↔ ((𝐻‘𝑥) ∈ (𝐻 “ 𝐶) ∧ (𝐻‘𝑥) ∈ (◡𝑆 “ {(𝐻‘𝐷)})))
28	25, 26, 27	3imtr4g 296	. . . . 5 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) → (𝐻‘𝑥) ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)}))))
29		n0i 4311	. . . . 5 ⊢ ((𝐻‘𝑥) ∈ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) → ¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅)
30	28, 29	syl6 35	. . . 4 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝑥 ∈ (𝐶 ∩ (◡𝑅 “ {𝐷})) → ¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))
31	30	exlimdv 1933	. . 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 242	. 2 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (¬ (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ → ¬ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅))
33	32	con4d 115	1 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅ → (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3455   ∩ cin 3921   ⊆ wss 3922  ∅c0 4304  {csn 4597   class class class wbr 5115  ^◡ccnv 5645   “ cima 5649   Fn wfn 6514  ‘cfv 6519   RelPres wrelp 44904

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-fv 6527  df-relp 44905

This theorem is referenced by:  relpfrlem  44915