Theorem wfrlem4OLD 7759

Description: Obsolete version of wfrlem4 7758 as of 18-Jul-2022. (Contributed by Scott Fenton, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
wfrlem4OLD.1	⊢ 𝑅 We 𝐴
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	wfrlem2 7755	. . . . 5 ⊢ (𝑔 ∈ 𝐵 → Fun 𝑔)
3		funfn 6216	. . . . 5 ⊢ (Fun 𝑔 ↔ 𝑔 Fn dom 𝑔)
4	2, 3	sylib 210	. . . 4 ⊢ (𝑔 ∈ 𝐵 → 𝑔 Fn dom 𝑔)
5		fnresin1 6302	. . . 4 ⊢ (𝑔 Fn dom 𝑔 → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ))
6	4, 5	syl 17	. . 3 ⊢ (𝑔 ∈ 𝐵 → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ))
7	6	adantr 473	. 2 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ))
8		inss1 4091	. . . . . . . 8 ⊢ (dom 𝑔 ∩ dom ℎ) ⊆ dom 𝑔
9	8	sseli 3853	. . . . . . 7 ⊢ (𝑎 ∈ (dom 𝑔 ∩ dom ℎ) → 𝑎 ∈ dom 𝑔)
10	1	wfrlem1 7754	. . . . . . . . 9 ⊢ 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))))}
11	10	abeq2i 2897	. . . . . . . 8 ⊢ (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)))))
12		fndm 6286	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝑏 → dom 𝑔 = 𝑏)
13	12	raleqdv 3352	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝑏 → (∀𝑎 ∈ dom 𝑔(𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ↔ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
14	13	biimpar 470	. . . . . . . . . . 11 ⊢ ((𝑔 Fn 𝑏 ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → ∀𝑎 ∈ dom 𝑔(𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
15		rsp 3152	. . . . . . . . . . 11 ⊢ (∀𝑎 ∈ dom 𝑔(𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) → (𝑎 ∈ dom 𝑔 → (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
16	14, 15	syl 17	. . . . . . . . . 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)))))
17	16	3adant2 1111	. . . . . . . . 9 ⊢ ((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)))))
18	17	exlimiv 1889	. . . . . . . 8 ⊢ (∃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)))))
19	11, 18	sylbi 209	. . . . . . 7 ⊢ (𝑔 ∈ 𝐵 → (𝑎 ∈ dom 𝑔 → (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
20	9, 19	syl5 34	. . . . . 6 ⊢ (𝑔 ∈ 𝐵 → (𝑎 ∈ (dom 𝑔 ∩ dom ℎ) → (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
21	20	imp 398	. . . . 5 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
22	21	adantlr 702	. . . 4 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
23		fvres 6516	. . . . 5 ⊢ (𝑎 ∈ (dom 𝑔 ∩ dom ℎ) → ((𝑔 ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑔‘𝑎))
24	23	adantl 474	. . . 4 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → ((𝑔 ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑔‘𝑎))
25		resres 5709	. . . . . 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)))
26		predss 5991	. . . . . . . . 9 ⊢ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)
27		sseqin2 4078	. . . . . . . . 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))
28	26, 27	mpbi 222	. . . . . . . 8 ⊢ ((dom 𝑔 ∩ dom ℎ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)) = Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)
29	1	wfrlem1 7754	. . . . . . . . . . . 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))))}
30	29	abeq2i 2897	. . . . . . . . . . 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)))))
31		3an6 1425	. . . . . . . . . . . . . 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))))))
32	31	2exbii 1811	. . . . . . . . . . . . 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))))))
33		eeanv 2281	. . . . . . . . . . . . 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))))))
34	32, 33	bitri 267	. . . . . . . . . . . 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))))))
35		ssinss1 4100	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ⊆ 𝐴 → (𝑏 ∩ 𝑐) ⊆ 𝐴)
36	35	ad2antrr 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → (𝑏 ∩ 𝑐) ⊆ 𝐴)
37		nfra1 3166	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑎∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏
38		nfra1 3166	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑎∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐
39	37, 38	nfan 1862	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑎(∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)
40		inss1 4091	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 ∩ 𝑐) ⊆ 𝑏
41	40	sseli 3853	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 ∈ (𝑏 ∩ 𝑐) → 𝑎 ∈ 𝑏)
42		rsp 3152	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 → (𝑎 ∈ 𝑏 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏))
43	41, 42	syl5com 31	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ (𝑏 ∩ 𝑐) → (∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏))
44		inss2 4092	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 ∩ 𝑐) ⊆ 𝑐
45	44	sseli 3853	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 ∈ (𝑏 ∩ 𝑐) → 𝑎 ∈ 𝑐)
46		rsp 3152	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 → (𝑎 ∈ 𝑐 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))
47	45, 46	syl5com 31	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ (𝑏 ∩ 𝑐) → (∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))
48	43, 47	anim12d 599	. . . . . . . . . . . . . . . . . . . 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)))
49		ssin 4093	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ↔ Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐))
50	49	biimpi 208	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐))
51	48, 50	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)))
52	39, 51	ralrimi 3163	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → ∀𝑎 ∈ (𝑏 ∩ 𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐))
53	52	ad2ant2l 733	. . . . . . . . . . . . . . . . 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))
54	36, 53	jca 504	. . . . . . . . . . . . . . . 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)))
55		fndm 6286	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ Fn 𝑐 → dom ℎ = 𝑐)
56	12, 55	ineqan12d 4077	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 Fn 𝑏 ∧ ℎ Fn 𝑐) → (dom 𝑔 ∩ dom ℎ) = (𝑏 ∩ 𝑐))
57		sseq1 3881	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom 𝑔 ∩ dom ℎ) = (𝑏 ∩ 𝑐) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ↔ (𝑏 ∩ 𝑐) ⊆ 𝐴))
58		sseq2 3882	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((dom 𝑔 ∩ dom ℎ) = (𝑏 ∩ 𝑐) → (Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ) ↔ Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐)))
59	58	raleqbi1dv 3340	. . . . . . . . . . . . . . . . . . 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)))
60	57, 59	anbi12d 621	. . . . . . . . . . . . . . . . . 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))))
61	60	imbi2d 333	. . . . . . . . . . . . . . . . 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)))))
62	56, 61	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)))))
63	54, 62	mpbiri 250	. . . . . . . . . . . . . . 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))))
64	63	imp 398	. . . . . . . . . . . . . 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)))
65	64	3adant3 1112	. . . . . . . . . . . . 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)))
66	65	exlimivv 1891	. . . . . . . . . . . 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)))
67	34, 66	sylbir 227	. . . . . . . . . . 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)))
68	11, 30, 67	syl2anb 588	. . . . . . . . . 10 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)))
69	68	adantr 473	. . . . . . . . 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)))
70		simpr 477	. . . . . . . . 9 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → 𝑎 ∈ (dom 𝑔 ∩ dom ℎ))
71		preddowncl 6011	. . . . . . . . 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)))
72	69, 70, 71	sylc 65	. . . . . . . 8 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎) = Pred(𝑅, 𝐴, 𝑎))
73	28, 72	syl5eq 2823	. . . . . . 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))
74	73	reseq2d 5692	. . . . . 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)))
75	25, 74	syl5eq 2823	. . . . 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)))
76	75	fveq2d 6501	. . . 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))))
77	22, 24, 76	3eqtr4d 2821	. . 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))))
78	77	ralrimiva 3129	. 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))))
79	7, 78	jca 504	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 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507  ∃wex 1742   ∈ wcel 2048  {cab 2755  ∀wral 3085   ∩ cin 3827   ⊆ wss 3828   We wwe 5362  dom cdm 5404   ↾ cres 5406  Predcpred 5983  Fun wfun 6180   Fn wfn 6181  ‘cfv 6186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-sep 5058  ax-nul 5065  ax-pr 5184

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ral 3090  df-rex 3091  df-rab 3094  df-v 3414  df-sbc 3681  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-nul 4178  df-if 4349  df-sn 4440  df-pr 4442  df-op 4446  df-uni 4711  df-br 4928  df-opab 4990  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-iota 6150  df-fun 6188  df-fn 6189  df-fv 6194

This theorem is referenced by: (None)