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Theorem frrlem12 8188
Description: Lemma for well-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 4100 . . . 4 (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ (𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 ∈ {𝑧}))
2 velsn 4594 . . . . 5 (𝑤 ∈ {𝑧} ↔ 𝑤 = 𝑧)
32orbi2i 911 . . . 4 ((𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 ∈ {𝑧}) ↔ (𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 = 𝑧))
41, 3bitri 275 . . 3 (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ (𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 = 𝑧))
5 elinel2 4148 . . . . . . . 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 8177 . . . . . . . . 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 5100 . . . . . . . . . . . . 13 (𝑥 = 𝑞 → (𝑥𝑔𝑢𝑞𝑔𝑢))
10 breq1 5100 . . . . . . . . . . . . 13 (𝑥 = 𝑞 → (𝑥𝑣𝑞𝑣))
119, 10anbi12d 632 . . . . . . . . . . . 12 (𝑥 = 𝑞 → ((𝑥𝑔𝑢𝑥𝑣) ↔ (𝑞𝑔𝑢𝑞𝑣)))
1211imbi1d 342 . . . . . . . . . . 11 (𝑥 = 𝑞 → (((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣) ↔ ((𝑞𝑔𝑢𝑞𝑣) → 𝑢 = 𝑣)))
1312imbi2d 341 . . . . . . . . . 10 (𝑥 = 𝑞 → (((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣)) ↔ ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑞𝑔𝑢𝑞𝑣) → 𝑢 = 𝑣))))
14 frrlem11.3 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
1513, 14chvarvv 2002 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑞𝑔𝑢𝑞𝑣) → 𝑢 = 𝑣))
167, 8, 15frrlem10 8186 . . . . . . . 8 ((𝜑𝑤 ∈ dom 𝐹) → (𝐹𝑤) = (𝑤𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑤))))
175, 16sylan2 594 . . . . . . 7 ((𝜑𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐹𝑤) = (𝑤𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑤))))
1817adantlr 713 . . . . . 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 6831 . . . . . . . 8 (𝐶𝑤) = (((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})‘𝑤)
216, 8, 14frrlem9 8185 . . . . . . . . . . . . 13 (𝜑 → Fun 𝐹)
2221funresd 6532 . . . . . . . . . . . 12 (𝜑 → Fun (𝐹𝑆))
23 dmres 5950 . . . . . . . . . . . 12 dom (𝐹𝑆) = (𝑆 ∩ dom 𝐹)
24 df-fn 6487 . . . . . . . . . . . 12 ((𝐹𝑆) Fn (𝑆 ∩ dom 𝐹) ↔ (Fun (𝐹𝑆) ∧ dom (𝐹𝑆) = (𝑆 ∩ dom 𝐹)))
2522, 23, 24sylanblrc 591 . . . . . . . . . . 11 (𝜑 → (𝐹𝑆) Fn (𝑆 ∩ dom 𝐹))
2625adantr 482 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝐹𝑆) Fn (𝑆 ∩ dom 𝐹))
2726adantr 482 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐹𝑆) Fn (𝑆 ∩ dom 𝐹))
28 vex 3446 . . . . . . . . . . 11 𝑧 ∈ V
29 ovex 7375 . . . . . . . . . . 11 (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
3028, 29fnsn 6547 . . . . . . . . . 10 {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧}
3130a1i 11 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧})
32 eldifn 4079 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
33 elinel2 4148 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑆 ∩ dom 𝐹) → 𝑧 ∈ dom 𝐹)
3432, 33nsyl 140 . . . . . . . . . . . 12 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ (𝑆 ∩ dom 𝐹))
35 disjsn 4664 . . . . . . . . . . . 12 (((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ (𝑆 ∩ dom 𝐹))
3634, 35sylibr 233 . . . . . . . . . . 11 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅)
3736adantl 483 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅)
3837adantr 482 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → ((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅)
39 simpr 486 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → 𝑤 ∈ (𝑆 ∩ dom 𝐹))
40 fvun1 6920 . . . . . . . . 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))
4127, 31, 38, 39, 40syl112anc 1374 . . . . . . . 8 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})‘𝑤) = ((𝐹𝑆)‘𝑤))
4220, 41eqtrid 2789 . . . . . . 7 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐶𝑤) = ((𝐹𝑆)‘𝑤))
43 elinel1 4147 . . . . . . . . 9 (𝑤 ∈ (𝑆 ∩ dom 𝐹) → 𝑤𝑆)
4443adantl 483 . . . . . . . 8 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → 𝑤𝑆)
4544fvresd 6850 . . . . . . 7 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → ((𝐹𝑆)‘𝑤) = (𝐹𝑤))
4642, 45eqtrd 2777 . . . . . 6 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐶𝑤) = (𝐹𝑤))
476, 8, 14, 19frrlem11 8187 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
48 fnfun 6590 . . . . . . . . . . 11 (𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → Fun 𝐶)
4947, 48syl 17 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → Fun 𝐶)
5049adantr 482 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → Fun 𝐶)
51 ssun1 4124 . . . . . . . . . . 11 (𝐹𝑆) ⊆ ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
5251, 19sseqtrri 3973 . . . . . . . . . 10 (𝐹𝑆) ⊆ 𝐶
5352a1i 11 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐹𝑆) ⊆ 𝐶)
54 eldifi 4078 . . . . . . . . . . . . 13 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧𝐴)
55 frrlem12.7 . . . . . . . . . . . . 13 ((𝜑𝑧𝐴) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
5654, 55sylan2 594 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
57 rspa 3228 . . . . . . . . . . . 12 ((∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆𝑤𝑆) → Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
5856, 43, 57syl2an 597 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
596, 8frrlem8 8184 . . . . . . . . . . . . 13 (𝑤 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
605, 59syl 17 . . . . . . . . . . . 12 (𝑤 ∈ (𝑆 ∩ dom 𝐹) → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
6160adantl 483 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
6258, 61ssind 4184 . . . . . . . . . 10 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹))
6362, 23sseqtrrdi 3987 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑤) ⊆ dom (𝐹𝑆))
64 fun2ssres 6534 . . . . . . . . 9 ((Fun 𝐶 ∧ (𝐹𝑆) ⊆ 𝐶 ∧ Pred(𝑅, 𝐴, 𝑤) ⊆ dom (𝐹𝑆)) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)) = ((𝐹𝑆) ↾ Pred(𝑅, 𝐴, 𝑤)))
6550, 53, 63, 64syl3anc 1371 . . . . . . . 8 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)) = ((𝐹𝑆) ↾ Pred(𝑅, 𝐴, 𝑤)))
6658resabs1d 5959 . . . . . . . 8 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → ((𝐹𝑆) ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑤)))
6765, 66eqtrd 2777 . . . . . . 7 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑤)))
6867oveq2d 7358 . . . . . 6 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝑤𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑤))))
6918, 46, 683eqtr4d 2787 . . . . 5 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ (𝑆 ∩ dom 𝐹)) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
7069ex 414 . . . 4 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝑤 ∈ (𝑆 ∩ dom 𝐹) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
7128, 29fvsn 7114 . . . . . 6 ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}‘𝑧) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
7219fveq1i 6831 . . . . . . 7 (𝐶𝑧) = (((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})‘𝑧)
7330a1i 11 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧})
74 vsnid 4615 . . . . . . . . 9 𝑧 ∈ {𝑧}
7574a1i 11 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑧 ∈ {𝑧})
76 fvun2 6921 . . . . . . . 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))
7726, 73, 37, 75, 76syl112anc 1374 . . . . . . 7 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})‘𝑧) = ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}‘𝑧))
7872, 77eqtrid 2789 . . . . . 6 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝐶𝑧) = ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}‘𝑧))
7919reseq1i 5924 . . . . . . . . 9 (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = (((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ↾ Pred(𝑅, 𝐴, 𝑧))
80 resundir 5943 . . . . . . . . 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)))
8179, 80eqtri 2765 . . . . . . . 8 (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = (((𝐹𝑆) ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧)))
82 frrlem12.6 . . . . . . . . . . . 12 ((𝜑𝑧𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
8354, 82sylan2 594 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
8483resabs1d 5959 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ((𝐹𝑆) ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
85 frrlem12.5 . . . . . . . . . . . . 13 (𝜑𝑅 Fr 𝐴)
86 predfrirr 6278 . . . . . . . . . . . . 13 (𝑅 Fr 𝐴 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧))
8785, 86syl 17 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧))
8887adantr 482 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧))
89 ressnop0 7086 . . . . . . . . . . 11 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧) → ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧)) = ∅)
9088, 89syl 17 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ({⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧)) = ∅)
9184, 90uneq12d 4116 . . . . . . . . 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)) ∪ ∅))
92 un0 4342 . . . . . . . . 9 ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
9391, 92eqtrdi 2793 . . . . . . . 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)))
9481, 93eqtrid 2789 . . . . . . 7 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
9594oveq2d 7358 . . . . . 6 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝑧𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
9671, 78, 953eqtr4a 2803 . . . . 5 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝐶𝑧) = (𝑧𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
97 fveq2 6830 . . . . . 6 (𝑤 = 𝑧 → (𝐶𝑤) = (𝐶𝑧))
98 id 22 . . . . . . 7 (𝑤 = 𝑧𝑤 = 𝑧)
99 predeq3 6247 . . . . . . . 8 (𝑤 = 𝑧 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑧))
10099reseq2d 5928 . . . . . . 7 (𝑤 = 𝑧 → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))
10198, 100oveq12d 7360 . . . . . 6 (𝑤 = 𝑧 → (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝑧𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
10297, 101eqeq12d 2753 . . . . 5 (𝑤 = 𝑧 → ((𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ (𝐶𝑧) = (𝑧𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))))
10396, 102syl5ibrcom 247 . . . 4 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝑤 = 𝑧 → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
10470, 103jaod 857 . . 3 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ((𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 = 𝑧) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
1054, 104biimtrid 241 . 2 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
1061053impia 1117 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 397  wo 845  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2714  wral 3062  cdif 3899  cun 3900  cin 3901  wss 3902  c0 4274  {csn 4578  cop 4584   class class class wbr 5097   Fr wfr 5577  dom cdm 5625  cres 5627  Predcpred 6242  Fun wfun 6478   Fn wfn 6479  cfv 6484  (class class class)co 7342  frecscfrecs 8171
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5248  ax-nul 5255  ax-pr 5377
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-iun 4948  df-br 5098  df-opab 5160  df-id 5523  df-fr 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-iota 6436  df-fun 6486  df-fn 6487  df-fv 6492  df-ov 7345  df-frecs 8172
This theorem is referenced by:  frrlem13  8189
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