MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wfrlem4 Structured version   Visualization version   GIF version

Theorem wfrlem4 7952
Description: Lemma for well-founded recursion. Properties of the restriction of an acceptable function to the domain of another one. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by AV, 18-Jul-2022.)
Hypothesis
Ref Expression
wfrlem4.2 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
Assertion
Ref Expression
wfrlem4 ((𝑔𝐵𝐵) → ((𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))
Distinct variable groups:   𝐴,𝑎,𝑓,𝑔,,𝑥,𝑦   𝐵,𝑎   𝐹,𝑎,𝑓,𝑔,,𝑥,𝑦   𝑅,𝑎,𝑓,𝑔,,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑓,𝑔,)

Proof of Theorem wfrlem4
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wfrlem4.2 . . . . . 6 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
21wfrlem2 7949 . . . . 5 (𝑔𝐵 → Fun 𝑔)
32funfnd 6382 . . . 4 (𝑔𝐵𝑔 Fn dom 𝑔)
4 fnresin1 6468 . . . 4 (𝑔 Fn dom 𝑔 → (𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ))
53, 4syl 17 . . 3 (𝑔𝐵 → (𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ))
65adantr 481 . 2 ((𝑔𝐵𝐵) → (𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ))
71wfrlem1 7948 . . . . . . . 8 𝐵 = {𝑔 ∣ ∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))}
87abeq2i 2952 . . . . . . 7 (𝑔𝐵 ↔ ∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
9 fndm 6451 . . . . . . . . . . . 12 (𝑔 Fn 𝑏 → dom 𝑔 = 𝑏)
109raleqdv 3420 . . . . . . . . . . 11 (𝑔 Fn 𝑏 → (∀𝑎 ∈ dom 𝑔(𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ↔ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
1110biimpar 478 . . . . . . . . . 10 ((𝑔 Fn 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → ∀𝑎 ∈ dom 𝑔(𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
12 rsp 3209 . . . . . . . . . 10 (∀𝑎 ∈ dom 𝑔(𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) → (𝑎 ∈ dom 𝑔 → (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
1311, 12syl 17 . . . . . . . . 9 ((𝑔 Fn 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → (𝑎 ∈ dom 𝑔 → (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
14133adant2 1125 . . . . . . . 8 ((𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → (𝑎 ∈ dom 𝑔 → (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
1514exlimiv 1924 . . . . . . 7 (∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → (𝑎 ∈ dom 𝑔 → (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
168, 15sylbi 218 . . . . . 6 (𝑔𝐵 → (𝑎 ∈ dom 𝑔 → (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
17 elinel1 4175 . . . . . 6 (𝑎 ∈ (dom 𝑔 ∩ dom ) → 𝑎 ∈ dom 𝑔)
1816, 17impel 506 . . . . 5 ((𝑔𝐵𝑎 ∈ (dom 𝑔 ∩ dom )) → (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
1918adantlr 711 . . . 4 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
20 fvres 6685 . . . . 5 (𝑎 ∈ (dom 𝑔 ∩ dom ) → ((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑔𝑎))
2120adantl 482 . . . 4 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑔𝑎))
22 resres 5864 . . . . . 6 ((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = (𝑔 ↾ ((dom 𝑔 ∩ dom ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))
23 predss 6152 . . . . . . . . 9 Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎) ⊆ (dom 𝑔 ∩ dom )
24 sseqin2 4195 . . . . . . . . 9 (Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎) ⊆ (dom 𝑔 ∩ dom ) ↔ ((dom 𝑔 ∩ dom ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))
2523, 24mpbi 231 . . . . . . . 8 ((dom 𝑔 ∩ dom ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)
261wfrlem1 7948 . . . . . . . . . . . 12 𝐵 = { ∣ ∃𝑐( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎))))}
2726abeq2i 2952 . . . . . . . . . . 11 (𝐵 ↔ ∃𝑐( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎)))))
28 3an6 1439 . . . . . . . . . . . . . 14 (((𝑔 Fn 𝑏 Fn 𝑐) ∧ ((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) ∧ (∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎))))) ↔ ((𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ ( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎))))))
29282exbii 1842 . . . . . . . . . . . . 13 (∃𝑏𝑐((𝑔 Fn 𝑏 Fn 𝑐) ∧ ((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) ∧ (∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎))))) ↔ ∃𝑏𝑐((𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ ( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎))))))
30 exdistrv 1949 . . . . . . . . . . . . 13 (∃𝑏𝑐((𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ ( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎))))) ↔ (∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ ∃𝑐( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎))))))
3129, 30bitri 276 . . . . . . . . . . . 12 (∃𝑏𝑐((𝑔 Fn 𝑏 Fn 𝑐) ∧ ((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) ∧ (∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎))))) ↔ (∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ ∃𝑐( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎))))))
32 ssinss1 4217 . . . . . . . . . . . . . . . . . 18 (𝑏𝐴 → (𝑏𝑐) ⊆ 𝐴)
3332ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → (𝑏𝑐) ⊆ 𝐴)
34 nfra1 3223 . . . . . . . . . . . . . . . . . . . 20 𝑎𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏
35 nfra1 3223 . . . . . . . . . . . . . . . . . . . 20 𝑎𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐
3634, 35nfan 1893 . . . . . . . . . . . . . . . . . . 19 𝑎(∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)
37 elinel1 4175 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (𝑏𝑐) → 𝑎𝑏)
38 rsp 3209 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 → (𝑎𝑏 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏))
3937, 38syl5com 31 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ (𝑏𝑐) → (∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏))
40 elinel2 4176 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (𝑏𝑐) → 𝑎𝑐)
41 rsp 3209 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 → (𝑎𝑐 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))
4240, 41syl5com 31 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ (𝑏𝑐) → (∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))
4339, 42anim12d 608 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ (𝑏𝑐) → ((∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → (Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)))
44 ssin 4210 . . . . . . . . . . . . . . . . . . . . 21 ((Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ↔ Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))
4544biimpi 217 . . . . . . . . . . . . . . . . . . . 20 ((Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))
4643, 45syl6com 37 . . . . . . . . . . . . . . . . . . 19 ((∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → (𝑎 ∈ (𝑏𝑐) → Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))
4736, 46ralrimi 3220 . . . . . . . . . . . . . . . . . 18 ((∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))
4847ad2ant2l 742 . . . . . . . . . . . . . . . . 17 (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))
4933, 48jca 512 . . . . . . . . . . . . . . . 16 (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ((𝑏𝑐) ⊆ 𝐴 ∧ ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))
50 fndm 6451 . . . . . . . . . . . . . . . . . 18 ( Fn 𝑐 → dom = 𝑐)
519, 50ineqan12d 4194 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn 𝑏 Fn 𝑐) → (dom 𝑔 ∩ dom ) = (𝑏𝑐))
52 sseq1 3995 . . . . . . . . . . . . . . . . . . 19 ((dom 𝑔 ∩ dom ) = (𝑏𝑐) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ↔ (𝑏𝑐) ⊆ 𝐴))
53 sseq2 3996 . . . . . . . . . . . . . . . . . . . 20 ((dom 𝑔 ∩ dom ) = (𝑏𝑐) → (Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ) ↔ Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))
5453raleqbi1dv 3408 . . . . . . . . . . . . . . . . . . 19 ((dom 𝑔 ∩ dom ) = (𝑏𝑐) → (∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ) ↔ ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))
5552, 54anbi12d 630 . . . . . . . . . . . . . . . . . 18 ((dom 𝑔 ∩ dom ) = (𝑏𝑐) → (((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )) ↔ ((𝑏𝑐) ⊆ 𝐴 ∧ ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))))
5655imbi2d 342 . . . . . . . . . . . . . . . . 17 ((dom 𝑔 ∩ dom ) = (𝑏𝑐) → ((((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ))) ↔ (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ((𝑏𝑐) ⊆ 𝐴 ∧ ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))))
5751, 56syl 17 . . . . . . . . . . . . . . . 16 ((𝑔 Fn 𝑏 Fn 𝑐) → ((((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ))) ↔ (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ((𝑏𝑐) ⊆ 𝐴 ∧ ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))))
5849, 57mpbiri 259 . . . . . . . . . . . . . . 15 ((𝑔 Fn 𝑏 Fn 𝑐) → (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ))))
5958imp 407 . . . . . . . . . . . . . 14 (((𝑔 Fn 𝑏 Fn 𝑐) ∧ ((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
60593adant3 1126 . . . . . . . . . . . . 13 (((𝑔 Fn 𝑏 Fn 𝑐) ∧ ((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) ∧ (∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎))))) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
6160exlimivv 1926 . . . . . . . . . . . 12 (∃𝑏𝑐((𝑔 Fn 𝑏 Fn 𝑐) ∧ ((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) ∧ (∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎))))) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
6231, 61sylbir 236 . . . . . . . . . . 11 ((∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ ∃𝑐( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎))))) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
638, 27, 62syl2anb 597 . . . . . . . . . 10 ((𝑔𝐵𝐵) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
6463adantr 481 . . . . . . . . 9 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
65 simpr 485 . . . . . . . . 9 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → 𝑎 ∈ (dom 𝑔 ∩ dom ))
66 preddowncl 6172 . . . . . . . . 9 (((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )) → (𝑎 ∈ (dom 𝑔 ∩ dom ) → Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎) = Pred(𝑅, 𝐴, 𝑎)))
6764, 65, 66sylc 65 . . . . . . . 8 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎) = Pred(𝑅, 𝐴, 𝑎))
6825, 67syl5eq 2872 . . . . . . 7 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((dom 𝑔 ∩ dom ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = Pred(𝑅, 𝐴, 𝑎))
6968reseq2d 5851 . . . . . 6 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → (𝑔 ↾ ((dom 𝑔 ∩ dom ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))
7022, 69syl5eq 2872 . . . . 5 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))
7170fveq2d 6670 . . . 4 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
7219, 21, 713eqtr4d 2870 . . 3 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))))
7372ralrimiva 3186 . 2 ((𝑔𝐵𝐵) → ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))))
746, 73jca 512 1 ((𝑔𝐵𝐵) → ((𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  {cab 2803  wral 3142  cin 3938  wss 3939  dom cdm 5553  cres 5555  Predcpred 6144   Fn wfn 6346  cfv 6351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-iota 6311  df-fun 6353  df-fn 6354  df-fv 6359
This theorem is referenced by:  wfrlem5  7953
  Copyright terms: Public domain W3C validator