Theorem wfrlem4OLD 8352

Description: Obsolete version as of 18-Nov-2024. Lemma for well-ordered recursion. Properties of the restriction of an acceptable function to the domain of another one. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by AV, 18-Jul-2022.)

Hypothesis

Ref	Expression
wfrlem4OLD.2	⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}

Assertion

Ref	Expression
wfrlem4OLD	⊢ ((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) = (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)))))

Distinct variable groups:   𝐴,𝑎,𝑓,𝑔,ℎ,𝑥,𝑦   𝐵,𝑎   𝐹,𝑎,𝑓,𝑔,ℎ,𝑥,𝑦   𝑅,𝑎,𝑓,𝑔,ℎ,𝑥,𝑦

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

Proof of Theorem wfrlem4OLD

Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wfrlem4OLD.2	. . . . . 6 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2	1	wfrlem2OLD 8349	. . . . 5 ⊢ (𝑔 ∈ 𝐵 → Fun 𝑔)
3	2	funfnd 6597	. . . 4 ⊢ (𝑔 ∈ 𝐵 → 𝑔 Fn dom 𝑔)
4		fnresin1 6693	. . . 4 ⊢ (𝑔 Fn dom 𝑔 → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ))
5	3, 4	syl 17	. . 3 ⊢ (𝑔 ∈ 𝐵 → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ))
6	5	adantr 480	. 2 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ))
7	1	wfrlem1OLD 8348	. . . . . . . 8 ⊢ 𝐵 = {𝑔 ∣ ∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))}
8	7	eqabri 2885	. . . . . . 7 ⊢ (𝑔 ∈ 𝐵 ↔ ∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
9		fndm 6671	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝑏 → dom 𝑔 = 𝑏)
10	9	raleqdv 3326	. . . . . . . . . . 11 ⊢ (𝑔 Fn 𝑏 → (∀𝑎 ∈ dom 𝑔(𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ↔ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
11	10	biimpar 477	. . . . . . . . . 10 ⊢ ((𝑔 Fn 𝑏 ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → ∀𝑎 ∈ dom 𝑔(𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
12		rsp 3247	. . . . . . . . . 10 ⊢ (∀𝑎 ∈ dom 𝑔(𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) → (𝑎 ∈ dom 𝑔 → (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
13	11, 12	syl 17	. . . . . . . . 9 ⊢ ((𝑔 Fn 𝑏 ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → (𝑎 ∈ dom 𝑔 → (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
14	13	3adant2 1132	. . . . . . . 8 ⊢ ((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → (𝑎 ∈ dom 𝑔 → (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
15	14	exlimiv 1930	. . . . . . 7 ⊢ (∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → (𝑎 ∈ dom 𝑔 → (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
16	8, 15	sylbi 217	. . . . . 6 ⊢ (𝑔 ∈ 𝐵 → (𝑎 ∈ dom 𝑔 → (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
17		elinel1 4201	. . . . . 6 ⊢ (𝑎 ∈ (dom 𝑔 ∩ dom ℎ) → 𝑎 ∈ dom 𝑔)
18	16, 17	impel 505	. . . . 5 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
19	18	adantlr 715	. . . 4 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
20		fvres 6925	. . . . 5 ⊢ (𝑎 ∈ (dom 𝑔 ∩ dom ℎ) → ((𝑔 ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑔‘𝑎))
21	20	adantl 481	. . . 4 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → ((𝑔 ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑔‘𝑎))
22		resres 6010	. . . . . 6 ⊢ ((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)) = (𝑔 ↾ ((dom 𝑔 ∩ dom ℎ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)))
23		predss 6329	. . . . . . . . 9 ⊢ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)
24		sseqin2 4223	. . . . . . . . 9 ⊢ (Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ) ↔ ((dom 𝑔 ∩ dom ℎ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)) = Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎))
25	23, 24	mpbi 230	. . . . . . . 8 ⊢ ((dom 𝑔 ∩ dom ℎ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)) = Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)
26	1	wfrlem1OLD 8348	. . . . . . . . . . . 12 ⊢ 𝐵 = {ℎ ∣ ∃𝑐(ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))}
27	26	eqabri 2885	. . . . . . . . . . 11 ⊢ (ℎ ∈ 𝐵 ↔ ∃𝑐(ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ Pred(𝑅, 𝐴, 𝑎)))))
28		3an6 1448	. . . . . . . . . . . . . 14 ⊢ (((𝑔 Fn 𝑏 ∧ ℎ Fn 𝑐) ∧ ((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) ∧ (∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) ↔ ((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))))
29	28	2exbii 1849	. . . . . . . . . . . . 13 ⊢ (∃𝑏∃𝑐((𝑔 Fn 𝑏 ∧ ℎ Fn 𝑐) ∧ ((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) ∧ (∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) ↔ ∃𝑏∃𝑐((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))))
30		exdistrv 1955	. . . . . . . . . . . . 13 ⊢ (∃𝑏∃𝑐((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) ↔ (∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ ∃𝑐(ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))))
31	29, 30	bitri 275	. . . . . . . . . . . 12 ⊢ (∃𝑏∃𝑐((𝑔 Fn 𝑏 ∧ ℎ Fn 𝑐) ∧ ((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) ∧ (∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) ↔ (∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ ∃𝑐(ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))))
32		ssinss1 4246	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ⊆ 𝐴 → (𝑏 ∩ 𝑐) ⊆ 𝐴)
33	32	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → (𝑏 ∩ 𝑐) ⊆ 𝐴)
34		nfra1 3284	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑎∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏
35		nfra1 3284	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑎∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐
36	34, 35	nfan 1899	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑎(∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)
37		elinel1 4201	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 ∈ (𝑏 ∩ 𝑐) → 𝑎 ∈ 𝑏)
38		rsp 3247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 → (𝑎 ∈ 𝑏 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏))
39	37, 38	syl5com 31	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ (𝑏 ∩ 𝑐) → (∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏))
40		elinel2 4202	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 ∈ (𝑏 ∩ 𝑐) → 𝑎 ∈ 𝑐)
41		rsp 3247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 → (𝑎 ∈ 𝑐 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))
42	40, 41	syl5com 31	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ (𝑏 ∩ 𝑐) → (∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))
43	39, 42	anim12d 609	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ (𝑏 ∩ 𝑐) → ((∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → (Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)))
44		ssin 4239	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ↔ Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐))
45	44	biimpi 216	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐))
46	43, 45	syl6com 37	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → (𝑎 ∈ (𝑏 ∩ 𝑐) → Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐)))
47	36, 46	ralrimi 3257	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → ∀𝑎 ∈ (𝑏 ∩ 𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐))
48	47	ad2ant2l 746	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ∀𝑎 ∈ (𝑏 ∩ 𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐))
49	33, 48	jca 511	. . . . . . . . . . . . . . . 16 ⊢ (((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ((𝑏 ∩ 𝑐) ⊆ 𝐴 ∧ ∀𝑎 ∈ (𝑏 ∩ 𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐)))
50		fndm 6671	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ Fn 𝑐 → dom ℎ = 𝑐)
51	9, 50	ineqan12d 4222	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 Fn 𝑏 ∧ ℎ Fn 𝑐) → (dom 𝑔 ∩ dom ℎ) = (𝑏 ∩ 𝑐))
52		sseq1 4009	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom 𝑔 ∩ dom ℎ) = (𝑏 ∩ 𝑐) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ↔ (𝑏 ∩ 𝑐) ⊆ 𝐴))
53		sseq2 4010	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((dom 𝑔 ∩ dom ℎ) = (𝑏 ∩ 𝑐) → (Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ) ↔ Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐)))
54	53	raleqbi1dv 3338	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom 𝑔 ∩ dom ℎ) = (𝑏 ∩ 𝑐) → (∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ) ↔ ∀𝑎 ∈ (𝑏 ∩ 𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐)))
55	52, 54	anbi12d 632	. . . . . . . . . . . . . . . . . 18 ⊢ ((dom 𝑔 ∩ dom ℎ) = (𝑏 ∩ 𝑐) → (((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)) ↔ ((𝑏 ∩ 𝑐) ⊆ 𝐴 ∧ ∀𝑎 ∈ (𝑏 ∩ 𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐))))
56	55	imbi2d 340	. . . . . . . . . . . . . . . . 17 ⊢ ((dom 𝑔 ∩ dom ℎ) = (𝑏 ∩ 𝑐) → ((((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ))) ↔ (((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ((𝑏 ∩ 𝑐) ⊆ 𝐴 ∧ ∀𝑎 ∈ (𝑏 ∩ 𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐)))))
57	51, 56	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 Fn 𝑏 ∧ ℎ Fn 𝑐) → ((((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ))) ↔ (((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ((𝑏 ∩ 𝑐) ⊆ 𝐴 ∧ ∀𝑎 ∈ (𝑏 ∩ 𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐)))))
58	49, 57	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 Fn 𝑏 ∧ ℎ Fn 𝑐) → (((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ))))
59	58	imp 406	. . . . . . . . . . . . . 14 ⊢ (((𝑔 Fn 𝑏 ∧ ℎ Fn 𝑐) ∧ ((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)))
60	59	3adant3 1133	. . . . . . . . . . . . 13 ⊢ (((𝑔 Fn 𝑏 ∧ ℎ Fn 𝑐) ∧ ((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) ∧ (∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)))
61	60	exlimivv 1932	. . . . . . . . . . . 12 ⊢ (∃𝑏∃𝑐((𝑔 Fn 𝑏 ∧ ℎ Fn 𝑐) ∧ ((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) ∧ (∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)))
62	31, 61	sylbir 235	. . . . . . . . . . 11 ⊢ ((∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ ∃𝑐(ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝐹‘(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)))
63	8, 27, 62	syl2anb 598	. . . . . . . . . 10 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)))
64	63	adantr 480	. . . . . . . . 9 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)))
65		simpr 484	. . . . . . . . 9 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → 𝑎 ∈ (dom 𝑔 ∩ dom ℎ))
66		preddowncl 6353	. . . . . . . . 9 ⊢ (((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)) → (𝑎 ∈ (dom 𝑔 ∩ dom ℎ) → Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎) = Pred(𝑅, 𝐴, 𝑎)))
67	64, 65, 66	sylc 65	. . . . . . . 8 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎) = Pred(𝑅, 𝐴, 𝑎))
68	25, 67	eqtrid 2789	. . . . . . 7 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → ((dom 𝑔 ∩ dom ℎ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)) = Pred(𝑅, 𝐴, 𝑎))
69	68	reseq2d 5997	. . . . . 6 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → (𝑔 ↾ ((dom 𝑔 ∩ dom ℎ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎))) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))
70	22, 69	eqtrid 2789	. . . . 5 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → ((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))
71	70	fveq2d 6910	. . . 4 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎))) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
72	19, 21, 71	3eqtr4d 2787	. . 3 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → ((𝑔 ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎))))
73	72	ralrimiva 3146	. 2 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)((𝑔 ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎))))
74	6, 73	jca 511	1 ⊢ ((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) = (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  ∀wral 3061   ∩ cin 3950   ⊆ wss 3951  dom cdm 5685   ↾ cres 5687  Predcpred 6320   Fn wfn 6556  ‘cfv 6561

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569

This theorem is referenced by:  wfrlem5OLD  8353