Theorem fprlem1 8242

Description: Lemma for well-founded recursion with a partial order. Two acceptable functions are compatible. (Contributed by Scott Fenton, 11-Sep-2023.)

Hypotheses

Ref	Expression
fprlem.1	⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
fprlem.2	⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
fprlem1	⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))

Distinct variable groups:   𝐴,𝑓,𝑥,𝑦,𝑔,ℎ,𝑢,𝑣   𝑅,𝑓,𝑥,𝑦,𝑔,ℎ,𝑢,𝑣   𝑓,𝐺,𝑥,𝑦,𝑔,ℎ,𝑢,𝑣

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔,ℎ)   𝐹(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔,ℎ)

Proof of Theorem fprlem1

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3444	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3444	. . . . 5 ⊢ 𝑢 ∈ V
3	1, 2	breldm 5857	. . . 4 ⊢ (𝑥𝑔𝑢 → 𝑥 ∈ dom 𝑔)
4		vex 3444	. . . . 5 ⊢ 𝑣 ∈ V
5	1, 4	breldm 5857	. . . 4 ⊢ (𝑥ℎ𝑣 → 𝑥 ∈ dom ℎ)
6		elin 3917	. . . . 5 ⊢ (𝑥 ∈ (dom 𝑔 ∩ dom ℎ) ↔ (𝑥 ∈ dom 𝑔 ∧ 𝑥 ∈ dom ℎ))
7	6	biimpri 228	. . . 4 ⊢ ((𝑥 ∈ dom 𝑔 ∧ 𝑥 ∈ dom ℎ) → 𝑥 ∈ (dom 𝑔 ∩ dom ℎ))
8	3, 5, 7	syl2an 596	. . 3 ⊢ ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑥 ∈ (dom 𝑔 ∩ dom ℎ))
9		id 22	. . 3 ⊢ ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣))
10	2	brresi 5947	. . . . 5 ⊢ (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑢 ↔ (𝑥 ∈ (dom 𝑔 ∩ dom ℎ) ∧ 𝑥𝑔𝑢))
11	4	brresi 5947	. . . . 5 ⊢ (𝑥(ℎ ↾ (dom 𝑔 ∩ dom ℎ))𝑣 ↔ (𝑥 ∈ (dom 𝑔 ∩ dom ℎ) ∧ 𝑥ℎ𝑣))
12	10, 11	anbi12i 628	. . . 4 ⊢ ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑢 ∧ 𝑥(ℎ ↾ (dom 𝑔 ∩ dom ℎ))𝑣) ↔ ((𝑥 ∈ (dom 𝑔 ∩ dom ℎ) ∧ 𝑥𝑔𝑢) ∧ (𝑥 ∈ (dom 𝑔 ∩ dom ℎ) ∧ 𝑥ℎ𝑣)))
13		an4 656	. . . 4 ⊢ (((𝑥 ∈ (dom 𝑔 ∩ dom ℎ) ∧ 𝑥𝑔𝑢) ∧ (𝑥 ∈ (dom 𝑔 ∩ dom ℎ) ∧ 𝑥ℎ𝑣)) ↔ ((𝑥 ∈ (dom 𝑔 ∩ dom ℎ) ∧ 𝑥 ∈ (dom 𝑔 ∩ dom ℎ)) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)))
14	12, 13	bitri 275	. . 3 ⊢ ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑢 ∧ 𝑥(ℎ ↾ (dom 𝑔 ∩ dom ℎ))𝑣) ↔ ((𝑥 ∈ (dom 𝑔 ∩ dom ℎ) ∧ 𝑥 ∈ (dom 𝑔 ∩ dom ℎ)) ∧ (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)))
15	8, 8, 9, 14	syl21anbrc 1345	. 2 ⊢ ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑢 ∧ 𝑥(ℎ ↾ (dom 𝑔 ∩ dom ℎ))𝑣))
16		inss2 4190	. . . . . . . . . 10 ⊢ (dom 𝑔 ∩ dom ℎ) ⊆ dom ℎ
17		fprlem.1	. . . . . . . . . . 11 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
18	17	frrlem3 8230	. . . . . . . . . 10 ⊢ (ℎ ∈ 𝐵 → dom ℎ ⊆ 𝐴)
19	16, 18	sstrid 3945	. . . . . . . . 9 ⊢ (ℎ ∈ 𝐵 → (dom 𝑔 ∩ dom ℎ) ⊆ 𝐴)
20	19	adantl 481	. . . . . . . 8 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → (dom 𝑔 ∩ dom ℎ) ⊆ 𝐴)
21	20	adantl 481	. . . . . . 7 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → (dom 𝑔 ∩ dom ℎ) ⊆ 𝐴)
22		simpl1 1192	. . . . . . 7 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → 𝑅 Fr 𝐴)
23		frss 5588	. . . . . . 7 ⊢ ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 → (𝑅 Fr 𝐴 → 𝑅 Fr (dom 𝑔 ∩ dom ℎ)))
24	21, 22, 23	sylc 65	. . . . . 6 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → 𝑅 Fr (dom 𝑔 ∩ dom ℎ))
25		simpl2 1193	. . . . . . 7 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → 𝑅 Po 𝐴)
26		poss 5534	. . . . . . 7 ⊢ ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 → (𝑅 Po 𝐴 → 𝑅 Po (dom 𝑔 ∩ dom ℎ)))
27	21, 25, 26	sylc 65	. . . . . 6 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → 𝑅 Po (dom 𝑔 ∩ dom ℎ))
28		simpl3 1194	. . . . . . 7 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → 𝑅 Se 𝐴)
29		sess2 5590	. . . . . . 7 ⊢ ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 → (𝑅 Se 𝐴 → 𝑅 Se (dom 𝑔 ∩ dom ℎ)))
30	21, 28, 29	sylc 65	. . . . . 6 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → 𝑅 Se (dom 𝑔 ∩ dom ℎ))
31	17	frrlem4 8231	. . . . . . 7 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → ((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)((𝑔 ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)))))
32	31	adantl 481	. . . . . 6 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)((𝑔 ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)))))
33	17	frrlem4 8231	. . . . . . . . 9 ⊢ ((ℎ ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → ((ℎ ↾ (dom ℎ ∩ dom 𝑔)) Fn (dom ℎ ∩ dom 𝑔) ∧ ∀𝑎 ∈ (dom ℎ ∩ dom 𝑔)((ℎ ↾ (dom ℎ ∩ dom 𝑔))‘𝑎) = (𝑎𝐺((ℎ ↾ (dom ℎ ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ℎ ∩ dom 𝑔), 𝑎)))))
34		incom 4161	. . . . . . . . . . . 12 ⊢ (dom 𝑔 ∩ dom ℎ) = (dom ℎ ∩ dom 𝑔)
35	34	reseq2i 5935	. . . . . . . . . . 11 ⊢ (ℎ ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom ℎ ∩ dom 𝑔))
36		fneq12 6588	. . . . . . . . . . 11 ⊢ (((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom ℎ ∩ dom 𝑔)) ∧ (dom 𝑔 ∩ dom ℎ) = (dom ℎ ∩ dom 𝑔)) → ((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ) ↔ (ℎ ↾ (dom ℎ ∩ dom 𝑔)) Fn (dom ℎ ∩ dom 𝑔)))
37	35, 34, 36	mp2an 692	. . . . . . . . . 10 ⊢ ((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ) ↔ (ℎ ↾ (dom ℎ ∩ dom 𝑔)) Fn (dom ℎ ∩ dom 𝑔))
38	35	fveq1i 6835	. . . . . . . . . . . 12 ⊢ ((ℎ ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = ((ℎ ↾ (dom ℎ ∩ dom 𝑔))‘𝑎)
39		predeq2 6262	. . . . . . . . . . . . . . 15 ⊢ ((dom 𝑔 ∩ dom ℎ) = (dom ℎ ∩ dom 𝑔) → Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎) = Pred(𝑅, (dom ℎ ∩ dom 𝑔), 𝑎))
40	34, 39	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎) = Pred(𝑅, (dom ℎ ∩ dom 𝑔), 𝑎)
41	35, 40	reseq12i 5936	. . . . . . . . . . . . 13 ⊢ ((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)) = ((ℎ ↾ (dom ℎ ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ℎ ∩ dom 𝑔), 𝑎))
42	41	oveq2i 7369	. . . . . . . . . . . 12 ⊢ (𝑎𝐺((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎))) = (𝑎𝐺((ℎ ↾ (dom ℎ ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ℎ ∩ dom 𝑔), 𝑎)))
43	38, 42	eqeq12i 2754	. . . . . . . . . . 11 ⊢ (((ℎ ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑎𝐺((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎))) ↔ ((ℎ ↾ (dom ℎ ∩ dom 𝑔))‘𝑎) = (𝑎𝐺((ℎ ↾ (dom ℎ ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ℎ ∩ dom 𝑔), 𝑎))))
44	34, 43	raleqbii 3314	. . . . . . . . . 10 ⊢ (∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)((ℎ ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑎𝐺((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎))) ↔ ∀𝑎 ∈ (dom ℎ ∩ dom 𝑔)((ℎ ↾ (dom ℎ ∩ dom 𝑔))‘𝑎) = (𝑎𝐺((ℎ ↾ (dom ℎ ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ℎ ∩ dom 𝑔), 𝑎))))
45	37, 44	anbi12i 628	. . . . . . . . 9 ⊢ (((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)((ℎ ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑎𝐺((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)))) ↔ ((ℎ ↾ (dom ℎ ∩ dom 𝑔)) Fn (dom ℎ ∩ dom 𝑔) ∧ ∀𝑎 ∈ (dom ℎ ∩ dom 𝑔)((ℎ ↾ (dom ℎ ∩ dom 𝑔))‘𝑎) = (𝑎𝐺((ℎ ↾ (dom ℎ ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ℎ ∩ dom 𝑔), 𝑎)))))
46	33, 45	sylibr 234	. . . . . . . 8 ⊢ ((ℎ ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → ((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)((ℎ ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑎𝐺((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)))))
47	46	ancoms 458	. . . . . . 7 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → ((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)((ℎ ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑎𝐺((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)))))
48	47	adantl 481	. . . . . 6 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)((ℎ ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑎𝐺((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)))))
49		fpr3g 8227	. . . . . 6 ⊢ (((𝑅 Fr (dom 𝑔 ∩ dom ℎ) ∧ 𝑅 Po (dom 𝑔 ∩ dom ℎ) ∧ 𝑅 Se (dom 𝑔 ∩ dom ℎ)) ∧ ((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)((𝑔 ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)))) ∧ ((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)((ℎ ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑎𝐺((ℎ ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎))))) → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
50	24, 27, 30, 32, 48, 49	syl311anc 1386	. . . . 5 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))
51	50	breqd 5109	. . . 4 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑣 ↔ 𝑥(ℎ ↾ (dom 𝑔 ∩ dom ℎ))𝑣))
52	51	biimprd 248	. . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → (𝑥(ℎ ↾ (dom 𝑔 ∩ dom ℎ))𝑣 → 𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑣))
53	17	frrlem2 8229	. . . . 5 ⊢ (𝑔 ∈ 𝐵 → Fun 𝑔)
54	53	ad2antrl 728	. . . 4 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → Fun 𝑔)
55		funres 6534	. . . 4 ⊢ (Fun 𝑔 → Fun (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)))
56		dffun2 6502	. . . . 5 ⊢ (Fun (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↔ (Rel (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ∧ ∀𝑥∀𝑢∀𝑣((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑢 ∧ 𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑣) → 𝑢 = 𝑣)))
57		2sp 2193	. . . . . 6 ⊢ (∀𝑢∀𝑣((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑢 ∧ 𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑣) → 𝑢 = 𝑣) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑢 ∧ 𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑣) → 𝑢 = 𝑣))
58	57	sps 2192	. . . . 5 ⊢ (∀𝑥∀𝑢∀𝑣((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑢 ∧ 𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑣) → 𝑢 = 𝑣) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑢 ∧ 𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑣) → 𝑢 = 𝑣))
59	56, 58	simplbiim 504	. . . 4 ⊢ (Fun (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑢 ∧ 𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑣) → 𝑢 = 𝑣))
60	54, 55, 59	3syl 18	. . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑢 ∧ 𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑣) → 𝑢 = 𝑣))
61	52, 60	sylan2d 605	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ℎ))𝑢 ∧ 𝑥(ℎ ↾ (dom 𝑔 ∩ dom ℎ))𝑣) → 𝑢 = 𝑣))
62	15, 61	syl5 34	1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714  ∀wral 3051   ∩ cin 3900   ⊆ wss 3901   class class class wbr 5098   Po wpo 5530   Fr wfr 5574   Se wse 5575  dom cdm 5624   ↾ cres 5626  Rel wrel 5629  Predcpred 6258  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358  frecscfrecs 8222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-fr 5577  df-se 5578  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-ov 7361

This theorem is referenced by:  fpr2a  8244  fpr1  8245  fprfung  8251