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Theorem wfrlem4OLD 7622
Description: Obsolete version of wfrlem4 7621 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 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
Assertion
Ref Expression
wfrlem4OLD ((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 wfrlem4OLD
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wfrlem4OLD.2 . . . . . 6 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
21wfrlem2 7618 . . . . 5 (𝑔𝐵 → Fun 𝑔)
3 funfn 6098 . . . . 5 (Fun 𝑔𝑔 Fn dom 𝑔)
42, 3sylib 209 . . . 4 (𝑔𝐵𝑔 Fn dom 𝑔)
5 fnresin1 6183 . . . 4 (𝑔 Fn dom 𝑔 → (𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ))
64, 5syl 17 . . 3 (𝑔𝐵 → (𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ))
76adantr 472 . 2 ((𝑔𝐵𝐵) → (𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ))
8 inss1 3992 . . . . . . . 8 (dom 𝑔 ∩ dom ) ⊆ dom 𝑔
98sseli 3757 . . . . . . 7 (𝑎 ∈ (dom 𝑔 ∩ dom ) → 𝑎 ∈ dom 𝑔)
101wfrlem1 7617 . . . . . . . . 9 𝐵 = {𝑔 ∣ ∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))}
1110abeq2i 2878 . . . . . . . 8 (𝑔𝐵 ↔ ∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
12 fndm 6168 . . . . . . . . . . . . 13 (𝑔 Fn 𝑏 → dom 𝑔 = 𝑏)
1312raleqdv 3292 . . . . . . . . . . . 12 (𝑔 Fn 𝑏 → (∀𝑎 ∈ dom 𝑔(𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) ↔ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
1413biimpar 469 . . . . . . . . . . 11 ((𝑔 Fn 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → ∀𝑎 ∈ dom 𝑔(𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
15 rsp 3076 . . . . . . . . . . 11 (∀𝑎 ∈ dom 𝑔(𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) → (𝑎 ∈ dom 𝑔 → (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
1614, 15syl 17 . . . . . . . . . 10 ((𝑔 Fn 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → (𝑎 ∈ dom 𝑔 → (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
17163adant2 1161 . . . . . . . . 9 ((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)))))
1817exlimiv 2025 . . . . . . . 8 (∃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)))))
1911, 18sylbi 208 . . . . . . 7 (𝑔𝐵 → (𝑎 ∈ dom 𝑔 → (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
209, 19syl5 34 . . . . . 6 (𝑔𝐵 → (𝑎 ∈ (dom 𝑔 ∩ dom ) → (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
2120imp 395 . . . . 5 ((𝑔𝐵𝑎 ∈ (dom 𝑔 ∩ dom )) → (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
2221adantlr 706 . . . 4 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → (𝑔𝑎) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
23 fvres 6394 . . . . 5 (𝑎 ∈ (dom 𝑔 ∩ dom ) → ((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑔𝑎))
2423adantl 473 . . . 4 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑔𝑎))
25 resres 5585 . . . . . 6 ((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = (𝑔 ↾ ((dom 𝑔 ∩ dom ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))
26 predss 5872 . . . . . . . . 9 Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎) ⊆ (dom 𝑔 ∩ dom )
27 sseqin2 3979 . . . . . . . . 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))
2826, 27mpbi 221 . . . . . . . 8 ((dom 𝑔 ∩ dom ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)
291wfrlem1 7617 . . . . . . . . . . . 12 𝐵 = { ∣ ∃𝑐( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎))))}
3029abeq2i 2878 . . . . . . . . . . 11 (𝐵 ↔ ∃𝑐( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ∧ ∀𝑎𝑐 (𝑎) = (𝐹‘( ↾ Pred(𝑅, 𝐴, 𝑎)))))
31 3an6 1570 . . . . . . . . . . . . . 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))))))
32312exbii 1944 . . . . . . . . . . . . 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))))))
33 eeanv 2346 . . . . . . . . . . . . 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))))))
3432, 33bitri 266 . . . . . . . . . . . 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))))))
35 ssinss1 4001 . . . . . . . . . . . . . . . . . 18 (𝑏𝐴 → (𝑏𝑐) ⊆ 𝐴)
3635ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → (𝑏𝑐) ⊆ 𝐴)
37 nfra1 3088 . . . . . . . . . . . . . . . . . . . 20 𝑎𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏
38 nfra1 3088 . . . . . . . . . . . . . . . . . . . 20 𝑎𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐
3937, 38nfan 1998 . . . . . . . . . . . . . . . . . . 19 𝑎(∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)
40 inss1 3992 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏𝑐) ⊆ 𝑏
4140sseli 3757 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (𝑏𝑐) → 𝑎𝑏)
42 rsp 3076 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 → (𝑎𝑏 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏))
4341, 42syl5com 31 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ (𝑏𝑐) → (∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏))
44 inss2 3993 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏𝑐) ⊆ 𝑐
4544sseli 3757 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (𝑏𝑐) → 𝑎𝑐)
46 rsp 3076 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 → (𝑎𝑐 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))
4745, 46syl5com 31 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ (𝑏𝑐) → (∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))
4843, 47anim12d 602 . . . . . . . . . . . . . . . . . . . 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)))
49 ssin 3994 . . . . . . . . . . . . . . . . . . . . 21 ((Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ↔ Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))
5049biimpi 207 . . . . . . . . . . . . . . . . . . . 20 ((Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))
5148, 50syl6com 37 . . . . . . . . . . . . . . . . . . 19 ((∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → (𝑎 ∈ (𝑏𝑐) → Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))
5239, 51ralrimi 3104 . . . . . . . . . . . . . . . . . 18 ((∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))
5352ad2ant2l 752 . . . . . . . . . . . . . . . . 17 (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))
5436, 53jca 507 . . . . . . . . . . . . . . . 16 (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)) → ((𝑏𝑐) ⊆ 𝐴 ∧ ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))
55 fndm 6168 . . . . . . . . . . . . . . . . . 18 ( Fn 𝑐 → dom = 𝑐)
5612, 55ineqan12d 3978 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn 𝑏 Fn 𝑐) → (dom 𝑔 ∩ dom ) = (𝑏𝑐))
57 sseq1 3786 . . . . . . . . . . . . . . . . . . 19 ((dom 𝑔 ∩ dom ) = (𝑏𝑐) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ↔ (𝑏𝑐) ⊆ 𝐴))
58 sseq2 3787 . . . . . . . . . . . . . . . . . . . 20 ((dom 𝑔 ∩ dom ) = (𝑏𝑐) → (Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ) ↔ Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))
5958raleqbi1dv 3294 . . . . . . . . . . . . . . . . . . 19 ((dom 𝑔 ∩ dom ) = (𝑏𝑐) → (∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ) ↔ ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))
6057, 59anbi12d 624 . . . . . . . . . . . . . . . . . 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))))
6160imbi2d 331 . . . . . . . . . . . . . . . . 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)))))
6256, 61syl 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)))))
6354, 62mpbiri 249 . . . . . . . . . . . . . . 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 ))))
6463imp 395 . . . . . . . . . . . . . 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 )))
65643adant3 1162 . . . . . . . . . . . . 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 )))
6665exlimivv 2027 . . . . . . . . . . . 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 )))
6734, 66sylbir 226 . . . . . . . . . . 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 )))
6811, 30, 67syl2anb 591 . . . . . . . . . 10 ((𝑔𝐵𝐵) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
6968adantr 472 . . . . . . . . 9 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
70 simpr 477 . . . . . . . . 9 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → 𝑎 ∈ (dom 𝑔 ∩ dom ))
71 preddowncl 5892 . . . . . . . . 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)))
7269, 70, 71sylc 65 . . . . . . . 8 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎) = Pred(𝑅, 𝐴, 𝑎))
7328, 72syl5eq 2811 . . . . . . 7 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((dom 𝑔 ∩ dom ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = Pred(𝑅, 𝐴, 𝑎))
7473reseq2d 5565 . . . . . 6 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → (𝑔 ↾ ((dom 𝑔 ∩ dom ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))
7525, 74syl5eq 2811 . . . . 5 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))
7675fveq2d 6379 . . . 4 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
7722, 24, 763eqtr4d 2809 . . 3 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))))
7877ralrimiva 3113 . 2 ((𝑔𝐵𝐵) → ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))))
797, 78jca 507 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 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wral 3055  cin 3731  wss 3732   We wwe 5235  dom cdm 5277  cres 5279  Predcpred 5864  Fun wfun 6062   Fn wfn 6063  cfv 6068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-iota 6031  df-fun 6070  df-fn 6071  df-fv 6076
This theorem is referenced by: (None)
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