Theorem relpfrlem 45493

Description: Lemma for relpfr 45494. Proved without using the Axiom of Replacement. This is isofrlem 7320 with weaker hypotheses. (Contributed by Eric Schmidt, 11-Oct-2025.)

Hypotheses

Ref	Expression
relpfrlem.1	⊢ (𝜑 → 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵))
relpfrlem.2	⊢ (𝜑 → (𝐻 “ 𝑥) ∈ V)

Assertion

Ref	Expression
relpfrlem	⊢ (𝜑 → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐻   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆

Proof of Theorem relpfrlem

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relpfrlem.1	. . . . . . 7 ⊢ (𝜑 → 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵))
2		relpf 45490	. . . . . . 7 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻:𝐴⟶𝐵)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
4		ffn 6687	. . . . . . . 8 ⊢ (𝐻:𝐴⟶𝐵 → 𝐻 Fn 𝐴)
5		n0 4305	. . . . . . . . . 10 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
6		fnfvima 7213	. . . . . . . . . . . . 13 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ⊆ 𝐴 ∧ 𝑦 ∈ 𝑥) → (𝐻‘𝑦) ∈ (𝐻 “ 𝑥))
7	6	ne0d 4294	. . . . . . . . . . . 12 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ⊆ 𝐴 ∧ 𝑦 ∈ 𝑥) → (𝐻 “ 𝑥) ≠ ∅)
8	7	3expia 1133	. . . . . . . . . . 11 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ⊆ 𝐴) → (𝑦 ∈ 𝑥 → (𝐻 “ 𝑥) ≠ ∅))
9	8	exlimdv 1952	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ⊆ 𝐴) → (∃𝑦 𝑦 ∈ 𝑥 → (𝐻 “ 𝑥) ≠ ∅))
10	5, 9	biimtrid 244	. . . . . . . . 9 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ⊆ 𝐴) → (𝑥 ≠ ∅ → (𝐻 “ 𝑥) ≠ ∅))
11	10	expimpd 457	. . . . . . . 8 ⊢ (𝐻 Fn 𝐴 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → (𝐻 “ 𝑥) ≠ ∅))
12	4, 11	syl 17	. . . . . . 7 ⊢ (𝐻:𝐴⟶𝐵 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → (𝐻 “ 𝑥) ≠ ∅))
13		fimass 6708	. . . . . . 7 ⊢ (𝐻:𝐴⟶𝐵 → (𝐻 “ 𝑥) ⊆ 𝐵)
14	12, 13	jctild 533	. . . . . 6 ⊢ (𝐻:𝐴⟶𝐵 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ((𝐻 “ 𝑥) ⊆ 𝐵 ∧ (𝐻 “ 𝑥) ≠ ∅)))
15	3, 14	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ((𝐻 “ 𝑥) ⊆ 𝐵 ∧ (𝐻 “ 𝑥) ≠ ∅)))
16		dffr3 6085	. . . . . 6 ⊢ (𝑆 Fr 𝐵 ↔ ∀𝑧((𝑧 ⊆ 𝐵 ∧ 𝑧 ≠ ∅) → ∃𝑤 ∈ 𝑧 (𝑧 ∩ (◡𝑆 “ {𝑤})) = ∅))
17		relpfrlem.2	. . . . . . 7 ⊢ (𝜑 → (𝐻 “ 𝑥) ∈ V)
18		sseq1 3961	. . . . . . . . . 10 ⊢ (𝑧 = (𝐻 “ 𝑥) → (𝑧 ⊆ 𝐵 ↔ (𝐻 “ 𝑥) ⊆ 𝐵))
19		neeq1 3018	. . . . . . . . . 10 ⊢ (𝑧 = (𝐻 “ 𝑥) → (𝑧 ≠ ∅ ↔ (𝐻 “ 𝑥) ≠ ∅))
20	18, 19	anbi12d 641	. . . . . . . . 9 ⊢ (𝑧 = (𝐻 “ 𝑥) → ((𝑧 ⊆ 𝐵 ∧ 𝑧 ≠ ∅) ↔ ((𝐻 “ 𝑥) ⊆ 𝐵 ∧ (𝐻 “ 𝑥) ≠ ∅)))
21		ineq1 4165	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐻 “ 𝑥) → (𝑧 ∩ (◡𝑆 “ {𝑤})) = ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})))
22	21	eqeq1d 2763	. . . . . . . . . 10 ⊢ (𝑧 = (𝐻 “ 𝑥) → ((𝑧 ∩ (◡𝑆 “ {𝑤})) = ∅ ↔ ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅))
23	22	rexeqbi1dv 3330	. . . . . . . . 9 ⊢ (𝑧 = (𝐻 “ 𝑥) → (∃𝑤 ∈ 𝑧 (𝑧 ∩ (◡𝑆 “ {𝑤})) = ∅ ↔ ∃𝑤 ∈ (𝐻 “ 𝑥)((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅))
24	20, 23	imbi12d 346	. . . . . . . 8 ⊢ (𝑧 = (𝐻 “ 𝑥) → (((𝑧 ⊆ 𝐵 ∧ 𝑧 ≠ ∅) → ∃𝑤 ∈ 𝑧 (𝑧 ∩ (◡𝑆 “ {𝑤})) = ∅) ↔ (((𝐻 “ 𝑥) ⊆ 𝐵 ∧ (𝐻 “ 𝑥) ≠ ∅) → ∃𝑤 ∈ (𝐻 “ 𝑥)((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅)))
25	24	spcgv 3555	. . . . . . 7 ⊢ ((𝐻 “ 𝑥) ∈ V → (∀𝑧((𝑧 ⊆ 𝐵 ∧ 𝑧 ≠ ∅) → ∃𝑤 ∈ 𝑧 (𝑧 ∩ (◡𝑆 “ {𝑤})) = ∅) → (((𝐻 “ 𝑥) ⊆ 𝐵 ∧ (𝐻 “ 𝑥) ≠ ∅) → ∃𝑤 ∈ (𝐻 “ 𝑥)((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅)))
26	17, 25	syl 17	. . . . . 6 ⊢ (𝜑 → (∀𝑧((𝑧 ⊆ 𝐵 ∧ 𝑧 ≠ ∅) → ∃𝑤 ∈ 𝑧 (𝑧 ∩ (◡𝑆 “ {𝑤})) = ∅) → (((𝐻 “ 𝑥) ⊆ 𝐵 ∧ (𝐻 “ 𝑥) ≠ ∅) → ∃𝑤 ∈ (𝐻 “ 𝑥)((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅)))
27	16, 26	biimtrid 244	. . . . 5 ⊢ (𝜑 → (𝑆 Fr 𝐵 → (((𝐻 “ 𝑥) ⊆ 𝐵 ∧ (𝐻 “ 𝑥) ≠ ∅) → ∃𝑤 ∈ (𝐻 “ 𝑥)((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅)))
28	15, 27	syl5d 73	. . . 4 ⊢ (𝜑 → (𝑆 Fr 𝐵 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑤 ∈ (𝐻 “ 𝑥)((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅)))
29	3	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → 𝐻:𝐴⟶𝐵)
30	29	ffund 6692	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → Fun 𝐻)
31		simpl 486	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (𝐻 “ 𝑥) ∧ ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅) → 𝑤 ∈ (𝐻 “ 𝑥))
32		fvelima 6928	. . . . . . . . . 10 ⊢ ((Fun 𝐻 ∧ 𝑤 ∈ (𝐻 “ 𝑥)) → ∃𝑦 ∈ 𝑥 (𝐻‘𝑦) = 𝑤)
33	30, 31, 32	syl2an 605	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐴) ∧ (𝑤 ∈ (𝐻 “ 𝑥) ∧ ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅)) → ∃𝑦 ∈ 𝑥 (𝐻‘𝑦) = 𝑤)
34		sneq 4591	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = (𝐻‘𝑦) → {𝑤} = {(𝐻‘𝑦)})
35	34	eqcoms 2769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐻‘𝑦) = 𝑤 → {𝑤} = {(𝐻‘𝑦)})
36	35	imaeq2d 6046	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐻‘𝑦) = 𝑤 → (◡𝑆 “ {𝑤}) = (◡𝑆 “ {(𝐻‘𝑦)}))
37	36	ineq2d 4172	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐻‘𝑦) = 𝑤 → ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {(𝐻‘𝑦)})))
38	37	eqeq1d 2763	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻‘𝑦) = 𝑤 → (((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅ ↔ ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {(𝐻‘𝑦)})) = ∅))
39	38	biimpd 231	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻‘𝑦) = 𝑤 → (((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅ → ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {(𝐻‘𝑦)})) = ∅))
40		ssel 3930	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ⊆ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
41	40	imdistani 576	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ∈ 𝑥) → (𝑥 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐴))
42		relpmin 45492	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐻 “ 𝑥) ∩ (◡𝑆 “ {(𝐻‘𝑦)})) = ∅ → (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅))
43	1, 41, 42	syl2an 605	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ∈ 𝑥)) → (((𝐻 “ 𝑥) ∩ (◡𝑆 “ {(𝐻‘𝑦)})) = ∅ → (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅))
44	39, 43	sylan9r 516	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ∈ 𝑥)) ∧ (𝐻‘𝑦) = 𝑤) → (((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅ → (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅))
45	44	adantld 494	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ∈ 𝑥)) ∧ (𝐻‘𝑦) = 𝑤) → ((𝑤 ∈ (𝐻 “ 𝑥) ∧ ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅))
46	45	exp42 439	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ⊆ 𝐴 → (𝑦 ∈ 𝑥 → ((𝐻‘𝑦) = 𝑤 → ((𝑤 ∈ (𝐻 “ 𝑥) ∧ ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)))))
47	46	imp 410	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (𝑦 ∈ 𝑥 → ((𝐻‘𝑦) = 𝑤 → ((𝑤 ∈ (𝐻 “ 𝑥) ∧ ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅))))
48	47	com3l 89	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑥 → ((𝐻‘𝑦) = 𝑤 → ((𝜑 ∧ 𝑥 ⊆ 𝐴) → ((𝑤 ∈ (𝐻 “ 𝑥) ∧ ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅) → (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅))))
49	48	com4t 93	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → ((𝑤 ∈ (𝐻 “ 𝑥) ∧ ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅) → (𝑦 ∈ 𝑥 → ((𝐻‘𝑦) = 𝑤 → (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅))))
50	49	imp 410	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐴) ∧ (𝑤 ∈ (𝐻 “ 𝑥) ∧ ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅)) → (𝑦 ∈ 𝑥 → ((𝐻‘𝑦) = 𝑤 → (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)))
51	50	reximdvai 3172	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐴) ∧ (𝑤 ∈ (𝐻 “ 𝑥) ∧ ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅)) → (∃𝑦 ∈ 𝑥 (𝐻‘𝑦) = 𝑤 → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅))
52	33, 51	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐴) ∧ (𝑤 ∈ (𝐻 “ 𝑥) ∧ ((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅)) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)
53	52	rexlimdvaa 3163	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∃𝑤 ∈ (𝐻 “ 𝑥)((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅ → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅))
54	53	ex 416	. . . . . 6 ⊢ (𝜑 → (𝑥 ⊆ 𝐴 → (∃𝑤 ∈ (𝐻 “ 𝑥)((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅ → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)))
55	54	adantrd 495	. . . . 5 ⊢ (𝜑 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → (∃𝑤 ∈ (𝐻 “ 𝑥)((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅ → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)))
56	55	a2d 29	. . . 4 ⊢ (𝜑 → (((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑤 ∈ (𝐻 “ 𝑥)((𝐻 “ 𝑥) ∩ (◡𝑆 “ {𝑤})) = ∅) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)))
57	28, 56	syld 47	. . 3 ⊢ (𝜑 → (𝑆 Fr 𝐵 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)))
58	57	alrimdv 1948	. 2 ⊢ (𝜑 → (𝑆 Fr 𝐵 → ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)))
59		dffr3 6085	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅))
60	58, 59	imbitrrdi 254	1 ⊢ (𝜑 → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085  Vcvv 3453   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4581   Fr wfr 5595  ^◡ccnv 5644   “ cima 5648  Fun wfun 6511   Fn wfn 6512  ⟶wf 6513  ‘cfv 6517   RelPres wrelp 45482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-fr 5598  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-relp 45483

This theorem is referenced by:  relpfr  45494