Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frrlem12 Structured version   Visualization version   GIF version

Theorem frrlem12 33247
Description: Lemma for founded recursion. Next, we calculate the value of 𝐶. (Contributed by Scott Fenton, 7-Dec-2022.)
Hypotheses
Ref Expression
frrlem11.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
frrlem11.2 𝐹 = frecs(𝑅, 𝐴, 𝐺)
frrlem11.3 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
frrlem11.4 𝐶 = ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
frrlem12.5 (𝜑𝑅 Fr 𝐴)
frrlem12.6 ((𝜑𝑧𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
frrlem12.7 ((𝜑𝑧𝐴) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
Assertion
Ref Expression
frrlem12 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦,𝑧   𝑓,𝐺,𝑥,𝑦,𝑧   𝑅,𝑓,𝑥,𝑦,𝑧   𝐵,𝑔,,𝑧   𝑥,𝐹,𝑢,𝑣,𝑧   𝜑,𝑓,𝑧   𝑓,𝐹   𝜑,𝑔,,𝑥,𝑢,𝑣   𝐴,,𝑤,𝑓,𝑦,𝑥   𝑤,𝐺   𝑤,𝑅   𝑦,𝐹   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑦,𝑤)   𝐴(𝑣,𝑢,𝑔)   𝐵(𝑦,𝑤,𝑣,𝑢,𝑓)   𝐶(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓,𝑔,)   𝑅(𝑣,𝑢,𝑔,)   𝑆(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓,𝑔,)   𝐹(𝑤,𝑔,)   𝐺(𝑣,𝑢,𝑔,)

Proof of Theorem frrlem12
Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elun 4076 . . . 4 (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ (𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 ∈ {𝑧}))
2 velsn 4541 . . . . 5 (𝑤 ∈ {𝑧} ↔ 𝑤 = 𝑧)
32orbi2i 910 . . . 4 ((𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 ∈ {𝑧}) ↔ (𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 = 𝑧))
41, 3bitri 278 . . 3 (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ (𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 = 𝑧))
5 elinel2 4123 . . . . . . . 8 (𝑤 ∈ (𝑆 ∩ dom 𝐹) → 𝑤 ∈ dom 𝐹)
6 frrlem11.1 . . . . . . . . . 10 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
76frrlem1 33236 . . . . . . . . 9 𝐵 = {𝑝 ∣ ∃𝑞(𝑝 Fn 𝑞 ∧ (𝑞𝐴 ∧ ∀𝑤𝑞 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑞) ∧ ∀𝑤𝑞 (𝑝𝑤) = (𝑤𝐺(𝑝 ↾ Pred(𝑅, 𝐴, 𝑤))))}
8 frrlem11.2 . . . . . . . . 9 𝐹 = frecs(𝑅, 𝐴, 𝐺)
9 breq1 5033 . . . . . . . . . . . . 13 (𝑥 = 𝑞 → (𝑥𝑔𝑢𝑞𝑔𝑢))
10 breq1 5033 . . . . . . . . . . . . 13 (𝑥 = 𝑞 → (𝑥𝑣𝑞𝑣))
119, 10anbi12d 633 . . . . . . . . . . . 12 (𝑥 = 𝑞 → ((𝑥𝑔𝑢𝑥𝑣) ↔ (𝑞𝑔𝑢𝑞𝑣)))
1211imbi1d 345 . . . . . . . . . . 11 (𝑥 = 𝑞 → (((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣) ↔ ((𝑞𝑔𝑢𝑞𝑣) → 𝑢 = 𝑣)))
1312imbi2d 344 . . . . . . . . . 10 (𝑥 = 𝑞 → (((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣)) ↔ ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑞𝑔𝑢𝑞𝑣) → 𝑢 = 𝑣))))
14 frrlem11.3 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
1513, 14chvarvv 2005 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑞𝑔𝑢𝑞𝑣) → 𝑢 = 𝑣))
167, 8, 15frrlem10 33245 . . . . . . . 8 ((𝜑𝑤 ∈ dom 𝐹) → (𝐹𝑤) = (𝑤𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑤))))
175, 16sylan2 595 . . . . . . 7 ((𝜑𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐹𝑤) = (𝑤𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑤))))
1817adantlr 714 . . . . . 6 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐹𝑤) = (𝑤𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑤))))
19 frrlem11.4 . . . . . . . . 9 𝐶 = ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
2019fveq1i 6646 . . . . . . . 8 (𝐶𝑤) = (((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})‘𝑤)
216, 8, 14frrlem9 33244 . . . . . . . . . . . . 13 (𝜑 → Fun 𝐹)
22 funres 6366 . . . . . . . . . . . . 13 (Fun 𝐹 → Fun (𝐹𝑆))
2321, 22syl 17 . . . . . . . . . . . 12 (𝜑 → Fun (𝐹𝑆))
24 dmres 5840 . . . . . . . . . . . 12 dom (𝐹𝑆) = (𝑆 ∩ dom 𝐹)
25 df-fn 6327 . . . . . . . . . . . 12 ((𝐹𝑆) Fn (𝑆 ∩ dom 𝐹) ↔ (Fun (𝐹𝑆) ∧ dom (𝐹𝑆) = (𝑆 ∩ dom 𝐹)))
2623, 24, 25sylanblrc 593 . . . . . . . . . . 11 (𝜑 → (𝐹𝑆) Fn (𝑆 ∩ dom 𝐹))
2726adantr 484 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝐹𝑆) Fn (𝑆 ∩ dom 𝐹))
2827adantr 484 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐹𝑆) Fn (𝑆 ∩ dom 𝐹))
29 vex 3444 . . . . . . . . . . 11 𝑧 ∈ V
30 ovex 7168 . . . . . . . . . . 11 (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
3129, 30fnsn 6382 . . . . . . . . . 10 {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧}
3231a1i 11 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧})
33 eldifn 4055 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
34 elinel2 4123 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑆 ∩ dom 𝐹) → 𝑧 ∈ dom 𝐹)
3533, 34nsyl 142 . . . . . . . . . . . 12 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ (𝑆 ∩ dom 𝐹))
36 disjsn 4607 . . . . . . . . . . . 12 (((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ (𝑆 ∩ dom 𝐹))
3735, 36sylibr 237 . . . . . . . . . . 11 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅)
3837adantl 485 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅)
3938adantr 484 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → ((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅)
40 simpr 488 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → 𝑤 ∈ (𝑆 ∩ dom 𝐹))
41 fvun1 6729 . . . . . . . . 9 (((𝐹𝑆) Fn (𝑆 ∩ dom 𝐹) ∧ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧} ∧ (((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅ ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹))) → (((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})‘𝑤) = ((𝐹𝑆)‘𝑤))
4228, 32, 39, 40, 41syl112anc 1371 . . . . . . . 8 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})‘𝑤) = ((𝐹𝑆)‘𝑤))
4320, 42syl5eq 2845 . . . . . . 7 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐶𝑤) = ((𝐹𝑆)‘𝑤))
44 elinel1 4122 . . . . . . . . 9 (𝑤 ∈ (𝑆 ∩ dom 𝐹) → 𝑤𝑆)
4544adantl 485 . . . . . . . 8 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → 𝑤𝑆)
4645fvresd 6665 . . . . . . 7 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → ((𝐹𝑆)‘𝑤) = (𝐹𝑤))
4743, 46eqtrd 2833 . . . . . 6 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐶𝑤) = (𝐹𝑤))
486, 8, 14, 19frrlem11 33246 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
49 fnfun 6423 . . . . . . . . . . 11 (𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → Fun 𝐶)
5048, 49syl 17 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → Fun 𝐶)
5150adantr 484 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → Fun 𝐶)
52 ssun1 4099 . . . . . . . . . . 11 (𝐹𝑆) ⊆ ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
5352, 19sseqtrri 3952 . . . . . . . . . 10 (𝐹𝑆) ⊆ 𝐶
5453a1i 11 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐹𝑆) ⊆ 𝐶)
55 eldifi 4054 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧𝐴)
56 frrlem12.7 . . . . . . . . . . . . 13 ((𝜑𝑧𝐴) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
5755, 56sylan2 595 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
58 rspa 3171 . . . . . . . . . . . 12 ((∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆𝑤𝑆) → Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
5957, 44, 58syl2an 598 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
606, 8frrlem8 33243 . . . . . . . . . . . . 13 (𝑤 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
615, 60syl 17 . . . . . . . . . . . 12 (𝑤 ∈ (𝑆 ∩ dom 𝐹) → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
6261adantl 485 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
6359, 62ssind 4159 . . . . . . . . . 10 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹))
6463, 24sseqtrrdi 3966 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑤) ⊆ dom (𝐹𝑆))
65 fun2ssres 6369 . . . . . . . . 9 ((Fun 𝐶 ∧ (𝐹𝑆) ⊆ 𝐶 ∧ Pred(𝑅, 𝐴, 𝑤) ⊆ dom (𝐹𝑆)) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)) = ((𝐹𝑆) ↾ Pred(𝑅, 𝐴, 𝑤)))
6651, 54, 64, 65syl3anc 1368 . . . . . . . 8 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)) = ((𝐹𝑆) ↾ Pred(𝑅, 𝐴, 𝑤)))
6759resabs1d 5849 . . . . . . . 8 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → ((𝐹𝑆) ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑤)))
6866, 67eqtrd 2833 . . . . . . 7 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑤)))
6968oveq2d 7151 . . . . . 6 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝑤𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑤))))
7018, 47, 693eqtr4d 2843 . . . . 5 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
7170ex 416 . . . 4 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝑤 ∈ (𝑆 ∩ dom 𝐹) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
7229, 30fvsn 6920 . . . . . 6 ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}‘𝑧) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
7319fveq1i 6646 . . . . . . 7 (𝐶𝑧) = (((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})‘𝑧)
7431a1i 11 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧})
75 vsnid 4562 . . . . . . . . 9 𝑧 ∈ {𝑧}
7675a1i 11 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑧 ∈ {𝑧})
77 fvun2 6730 . . . . . . . 8 (((𝐹𝑆) Fn (𝑆 ∩ dom 𝐹) ∧ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧} ∧ (((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧})) → (((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})‘𝑧) = ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}‘𝑧))
7827, 74, 38, 76, 77syl112anc 1371 . . . . . . 7 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})‘𝑧) = ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}‘𝑧))
7973, 78syl5eq 2845 . . . . . 6 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝐶𝑧) = ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}‘𝑧))
8019reseq1i 5814 . . . . . . . . 9 (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = (((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ↾ Pred(𝑅, 𝐴, 𝑧))
81 resundir 5833 . . . . . . . . 9 (((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ↾ Pred(𝑅, 𝐴, 𝑧)) = (((𝐹𝑆) ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧)))
8280, 81eqtri 2821 . . . . . . . 8 (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = (((𝐹𝑆) ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧)))
83 frrlem12.6 . . . . . . . . . . . 12 ((𝜑𝑧𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
8455, 83sylan2 595 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
8584resabs1d 5849 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ((𝐹𝑆) ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
86 frrlem12.5 . . . . . . . . . . . . 13 (𝜑𝑅 Fr 𝐴)
87 predfrirr 6145 . . . . . . . . . . . . 13 (𝑅 Fr 𝐴 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧))
8886, 87syl 17 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧))
8988adantr 484 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧))
90 ressnop0 6892 . . . . . . . . . . 11 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧) → ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧)) = ∅)
9189, 90syl 17 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧)) = ∅)
9285, 91uneq12d 4091 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (((𝐹𝑆) ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧))) = ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅))
93 un0 4298 . . . . . . . . 9 ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
9492, 93eqtrdi 2849 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (((𝐹𝑆) ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
9582, 94syl5eq 2845 . . . . . . 7 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
9695oveq2d 7151 . . . . . 6 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝑧𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
9772, 79, 963eqtr4a 2859 . . . . 5 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝐶𝑧) = (𝑧𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
98 fveq2 6645 . . . . . 6 (𝑤 = 𝑧 → (𝐶𝑤) = (𝐶𝑧))
99 id 22 . . . . . . 7 (𝑤 = 𝑧𝑤 = 𝑧)
100 predeq3 6120 . . . . . . . 8 (𝑤 = 𝑧 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑧))
101100reseq2d 5818 . . . . . . 7 (𝑤 = 𝑧 → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))
10299, 101oveq12d 7153 . . . . . 6 (𝑤 = 𝑧 → (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝑧𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
10398, 102eqeq12d 2814 . . . . 5 (𝑤 = 𝑧 → ((𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ (𝐶𝑧) = (𝑧𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))))
10497, 103syl5ibrcom 250 . . . 4 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝑤 = 𝑧 → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
10571, 104jaod 856 . . 3 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ((𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 = 𝑧) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
1064, 105syl5bi 245 . 2 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
1071063impia 1114 1 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wex 1781  wcel 2111  {cab 2776  wral 3106  cdif 3878  cun 3879  cin 3880  wss 3881  c0 4243  {csn 4525  cop 4531   class class class wbr 5030   Fr wfr 5475  dom cdm 5519  cres 5521  Predcpred 6115  Fun wfun 6318   Fn wfn 6319  cfv 6324  (class class class)co 7135  frecscfrecs 33230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-id 5425  df-fr 5478  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-ov 7138  df-frecs 33231
This theorem is referenced by:  frrlem13  33248
  Copyright terms: Public domain W3C validator