frrlem4 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > frrlem4		Structured version Visualization version GIF version

Theorem frrlem4 8307

Description: Lemma for well-founded recursion. Properties of the restriction of an acceptable function to the domain of another acceptable function. (Contributed by Paul Chapman, 21-Apr-2012.)

**Hypothesis**
Ref	Expression
frrlem4.1	⊢ 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))))}

**Assertion**
Ref	Expression
frrlem4	⊢ ((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)))))

Distinct variable groups: 𝐴,𝑎,𝑓,𝑔 𝐴,ℎ,𝑥,𝑦,𝑎 𝐵,𝑎 𝑓,ℎ,𝑥,𝑦 𝐺,𝑎,𝑓,𝑔 ℎ,𝐺,𝑥,𝑦 𝑥,𝑔,𝑦 𝑅,𝑎,𝑓,𝑔 𝑅,ℎ,𝑥,𝑦 Allowed substitution hints: 𝐵(𝑥, 𝑦, 𝑓, 𝑔, ℎ)

Proof of Theorem frrlem4 Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frrlem4.1	. . . . . 6 ⊢ 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))))}
2	1	frrlem2 8305	. . . . 5 ⊢ (𝑔 ∈ 𝐵 → Fun 𝑔)
3	2	funfnd 6571	. . . 4 ⊢ (𝑔 ∈ 𝐵 → 𝑔 Fn dom 𝑔)
4		fnresin1 6664	. . . 4 ⊢ (𝑔 Fn dom 𝑔 → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ))
5	3, 4	syl 18	. . 3 ⊢ (𝑔 ∈ 𝐵 → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ))
6	5	adantr 486	. 2 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ))
7	1	frrlem1 8304	. . . . . . . 8 ⊢ 𝐵 = {𝑔 ∣ ∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))}
8	7	eqabri 2903	. . . . . . 7 ⊢ (𝑔 ∈ 𝐵 ↔ ∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
9		fndm 6642	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝑏 → dom 𝑔 = 𝑏)
10	9	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏)) → dom 𝑔 = 𝑏)
11	10	raleqdv 3320	. . . . . . . . . 10 ⊢ ((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏)) → (∀𝑎 ∈ dom 𝑔(𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ↔ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
12	11	biimp3ar 1499	. . . . . . . . 9 ⊢ ((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → ∀𝑎 ∈ dom 𝑔(𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
13		rsp 3251	. . . . . . . . 9 ⊢ (∀𝑎 ∈ dom 𝑔(𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) → (𝑎 ∈ dom 𝑔 → (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
14	12, 13	syl 18	. . . . . . . 8 ⊢ ((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → (𝑎 ∈ dom 𝑔 → (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
15	14	exlimiv 1963	. . . . . . 7 ⊢ (∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → (𝑎 ∈ dom 𝑔 → (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
16	8, 15	sylbi 220	. . . . . 6 ⊢ (𝑔 ∈ 𝐵 → (𝑎 ∈ dom 𝑔 → (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
17		elinel1 4147	. . . . . 6 ⊢ (𝑎 ∈ (dom 𝑔 ∩ dom ℎ) → 𝑎 ∈ dom 𝑔)
18	16, 17	impel 515	. . . . 5 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
19	18	adantlr 728	. . . 4 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
20		simpr 490	. . . . 5 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → 𝑎 ∈ (dom 𝑔 ∩ dom ℎ))
21	20	fvresd 6905	. . . 4 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → ((𝑔 ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑔‘𝑎))
22		resres 5983	. . . . . 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 6312	. . . . . . . . 9 ⊢ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)
24		sseqin2 4169	. . . . . . . . 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 233	. . . . . . . 8 ⊢ ((dom 𝑔 ∩ dom ℎ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)) = Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)
26	1	frrlem1 8304	. . . . . . . . . . . 12 ⊢ 𝐵 = {ℎ ∣ ∃𝑐(ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))}
27	26	eqabri 2903	. . . . . . . . . . 11 ⊢ (ℎ ∈ 𝐵 ↔ ∃𝑐(ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎)))))
28		exdistrv 1988	. . . . . . . . . . . 12 ⊢ (∃𝑏∃𝑐((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) ↔ (∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ ∃𝑐(ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))))
29		inss1 4182	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∩ 𝑐) ⊆ 𝑏
30		simpl2l 1245	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → 𝑏 ⊆ 𝐴)
31	29, 30	sstrid 3942	. . . . . . . . . . . . . 14 ⊢ (((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → (𝑏 ∩ 𝑐) ⊆ 𝐴)
32		simp2r 1219	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏)
33		simp2r 1219	. . . . . . . . . . . . . . 15 ⊢ ((ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎)))) → ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)
34		nfra1 3287	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏
35		nfra1 3287	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐
36	34, 35	nfan 1932	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑎(∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)
37		elinel1 4147	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ (𝑏 ∩ 𝑐) → 𝑎 ∈ 𝑏)
38		rsp 3251	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 → (𝑎 ∈ 𝑏 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏))
39	37, 38	syl5com 32	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ (𝑏 ∩ 𝑐) → (∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏))
40		elinel2 4148	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ (𝑏 ∩ 𝑐) → 𝑎 ∈ 𝑐)
41		rsp 3251	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 → (𝑎 ∈ 𝑐 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))
42	40, 41	syl5com 32	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ (𝑏 ∩ 𝑐) → (∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))
43	39, 42	anim12d 621	. . . . . . . . . . . . . . . . 17 ⊢ (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 4184	. . . . . . . . . . . . . . . . . 18 ⊢ ((Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ↔ Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐))
45	44	biimpi 219	. . . . . . . . . . . . . . . . 17 ⊢ ((Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐))
46	43, 45	syl6com 38	. . . . . . . . . . . . . . . 16 ⊢ ((∀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 3261	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → ∀𝑎 ∈ (𝑏 ∩ 𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐))
48	32, 33, 47	syl2an 608	. . . . . . . . . . . . . 14 ⊢ (((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → ∀𝑎 ∈ (𝑏 ∩ 𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐))
49		simpl1 1210	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → 𝑔 Fn 𝑏)
50	49	fndmd 6644	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → dom 𝑔 = 𝑏)
51		simpr1 1213	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → ℎ Fn 𝑐)
52	51	fndmd 6644	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → dom ℎ = 𝑐)
53		ineq12 4161	. . . . . . . . . . . . . . . . 17 ⊢ ((dom 𝑔 = 𝑏 ∧ dom ℎ = 𝑐) → (dom 𝑔 ∩ dom ℎ) = (𝑏 ∩ 𝑐))
54	53	sseq1d 3962	. . . . . . . . . . . . . . . 16 ⊢ ((dom 𝑔 = 𝑏 ∧ dom ℎ = 𝑐) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ↔ (𝑏 ∩ 𝑐) ⊆ 𝐴))
55	53	sseq2d 3963	. . . . . . . . . . . . . . . . 17 ⊢ ((dom 𝑔 = 𝑏 ∧ dom ℎ = 𝑐) → (Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ) ↔ Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐)))
56	53, 55	raleqbidv 3335	. . . . . . . . . . . . . . . 16 ⊢ ((dom 𝑔 = 𝑏 ∧ dom ℎ = 𝑐) → (∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ) ↔ ∀𝑎 ∈ (𝑏 ∩ 𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏 ∩ 𝑐)))
57	54, 56	anbi12d 644	. . . . . . . . . . . . . . 15 ⊢ ((dom 𝑔 = 𝑏 ∧ dom ℎ = 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))))
58	50, 52, 57	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → (((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))))
59	31, 48, 58	mpbir2and 726	. . . . . . . . . . . . 13 ⊢ (((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)))
60	59	exlimivv 1965	. . . . . . . . . . . 12 ⊢ (∃𝑏∃𝑐((𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)))
61	28, 60	sylbir 238	. . . . . . . . . . 11 ⊢ ((∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎 ∈ 𝑏 (𝑔‘𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ ∃𝑐(ℎ Fn 𝑐 ∧ (𝑐 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎 ∈ 𝑐 (ℎ‘𝑎) = (𝑎𝐺(ℎ ↾ Pred(𝑅, 𝐴, 𝑎))))) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)))
62	8, 27, 61	syl2anb 610	. . . . . . . . . 10 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → ((dom 𝑔 ∩ dom ℎ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ℎ)))
63	62	adantr 486	. . . . . . . . 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)))
64		preddowncl 6335	. . . . . . . . 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)))
65	63, 20, 64	sylc 66	. . . . . . . 8 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎) = Pred(𝑅, 𝐴, 𝑎))
66	25, 65	eqtrid 2808	. . . . . . 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))
67	66	reseq2d 5970	. . . . . 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)))
68	22, 67	eqtrid 2808	. . . . 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)))
69	68	oveq2d 7436	. . . 4 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎))) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
70	19, 21, 69	3eqtr4d 2806	. . 3 ⊢ (((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom ℎ)) → ((𝑔 ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎))))
71	70	ralrimiva 3155	. 2 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)((𝑔 ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎))))
72	6, 71	jca 521	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) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2145 {cab 2739 ∀wral 3077 ∩ cin 3898 ⊆ wss 3899 dom cdm 5651 ↾ cres 5653 Predcpred 6303 Fn wfn 6533 ‘cfv 6538 (class class class)co 7420

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-pr 5391

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-clab 2740 df-cleq 2753 df-clel 2836 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-iota 6494 df-fun 6540 df-fn 6541 df-fv 6546 df-ov 7423

This theorem is used by: fprlem1 8318 frrlem15 9761

W3C validator