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Theorem frrlem4 31476
Description: Lemma for 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 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
Assertion
Ref Expression
frrlem4 ((𝑔𝐵𝐵) → ((𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))
Distinct variable groups:   𝐴,𝑎,𝑓,𝑔   𝐴,,𝑥,𝑦,𝑎   𝐵,𝑎   𝑓,,𝑥,𝑦   𝐺,𝑎,𝑓,𝑔   ,𝐺,𝑥,𝑦   𝑥,𝑔,𝑦   𝑅,𝑎,𝑓,𝑔   𝑅,,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑓,𝑔,)

Proof of Theorem frrlem4
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frrlem4.1 . . . . . 6 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
21frrlem2 31474 . . . . 5 (𝑔𝐵 → Fun 𝑔)
3 funfn 5879 . . . . 5 (Fun 𝑔𝑔 Fn dom 𝑔)
42, 3sylib 208 . . . 4 (𝑔𝐵𝑔 Fn dom 𝑔)
5 fnresin1 5965 . . . 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 481 . 2 ((𝑔𝐵𝐵) → (𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ))
8 inss1 3816 . . . . . . . 8 (dom 𝑔 ∩ dom ) ⊆ dom 𝑔
98sseli 3584 . . . . . . 7 (𝑎 ∈ (dom 𝑔 ∩ dom ) → 𝑎 ∈ dom 𝑔)
101frrlem1 31473 . . . . . . . . 9 𝐵 = {𝑔 ∣ ∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))}
1110abeq2i 2738 . . . . . . . 8 (𝑔𝐵 ↔ ∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))))
12 fndm 5950 . . . . . . . . . 10 (𝑔 Fn 𝑏 → dom 𝑔 = 𝑏)
13 rsp 2929 . . . . . . . . . . . 12 (∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))) → (𝑎𝑏 → (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
14133ad2ant3 1082 . . . . . . . . . . 11 ((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → (𝑎𝑏 → (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
15 eleq2 2693 . . . . . . . . . . . 12 (dom 𝑔 = 𝑏 → (𝑎 ∈ dom 𝑔𝑎𝑏))
1615imbi1d 331 . . . . . . . . . . 11 (dom 𝑔 = 𝑏 → ((𝑎 ∈ dom 𝑔 → (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ↔ (𝑎𝑏 → (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))))
1714, 16syl5ibrcom 237 . . . . . . . . . 10 ((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) → (dom 𝑔 = 𝑏 → (𝑎 ∈ dom 𝑔 → (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))))
1812, 17mpan9 486 . . . . . . . . 9 ((𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))) → (𝑎 ∈ dom 𝑔 → (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
1918exlimiv 1860 . . . . . . . 8 (∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))) → (𝑎 ∈ dom 𝑔 → (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
2011, 19sylbi 207 . . . . . . 7 (𝑔𝐵 → (𝑎 ∈ dom 𝑔 → (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
219, 20syl5 34 . . . . . 6 (𝑔𝐵 → (𝑎 ∈ (dom 𝑔 ∩ dom ) → (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))))
2221imp 445 . . . . 5 ((𝑔𝐵𝑎 ∈ (dom 𝑔 ∩ dom )) → (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
2322adantlr 750 . . . 4 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
24 fvres 6165 . . . . 5 (𝑎 ∈ (dom 𝑔 ∩ dom ) → ((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑔𝑎))
2524adantl 482 . . . 4 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑔𝑎))
26 resres 5372 . . . . . 6 ((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = (𝑔 ↾ ((dom 𝑔 ∩ dom ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))
27 predss 5649 . . . . . . . . 9 Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎) ⊆ (dom 𝑔 ∩ dom )
28 sseqin2 3800 . . . . . . . . 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))
2927, 28mpbi 220 . . . . . . . 8 ((dom 𝑔 ∩ dom ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)
301frrlem1 31473 . . . . . . . . . . . 12 𝐵 = { ∣ ∃𝑐( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎)))))}
3130abeq2i 2738 . . . . . . . . . . 11 (𝐵 ↔ ∃𝑐( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎))))))
32 eeanv 2186 . . . . . . . . . . . 12 (∃𝑏𝑐((𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))) ∧ ( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎)))))) ↔ (∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))) ∧ ∃𝑐( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎)))))))
33 simpl1 1062 . . . . . . . . . . . . . . . . 17 (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎))))) → 𝑏𝐴)
34 ssinss1 3824 . . . . . . . . . . . . . . . . 17 (𝑏𝐴 → (𝑏𝑐) ⊆ 𝐴)
3533, 34syl 17 . . . . . . . . . . . . . . . 16 (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎))))) → (𝑏𝑐) ⊆ 𝐴)
36 simpl2 1063 . . . . . . . . . . . . . . . . 17 (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎))))) → ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏)
37 simpr2 1066 . . . . . . . . . . . . . . . . 17 (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎))))) → ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)
38 nfra1 2941 . . . . . . . . . . . . . . . . . . 19 𝑎𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏
39 nfra1 2941 . . . . . . . . . . . . . . . . . . 19 𝑎𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐
4038, 39nfan 1830 . . . . . . . . . . . . . . . . . 18 𝑎(∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)
41 inss1 3816 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏𝑐) ⊆ 𝑏
4241sseli 3584 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ (𝑏𝑐) → 𝑎𝑏)
43 rsp 2929 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 → (𝑎𝑏 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏))
4442, 43syl5com 31 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ (𝑏𝑐) → (∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏))
45 inss2 3817 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏𝑐) ⊆ 𝑐
4645sseli 3584 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ (𝑏𝑐) → 𝑎𝑐)
47 rsp 2929 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 → (𝑎𝑐 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))
4846, 47syl5com 31 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ (𝑏𝑐) → (∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 → Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐))
4944, 48anim12d 585 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ (𝑏𝑐) → ((∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → (Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐)))
50 ssin 3818 . . . . . . . . . . . . . . . . . . . 20 ((Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) ↔ Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))
5150biimpi 206 . . . . . . . . . . . . . . . . . . 19 ((Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))
5249, 51syl6com 37 . . . . . . . . . . . . . . . . . 18 ((∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → (𝑎 ∈ (𝑏𝑐) → Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))
5340, 52ralrimi 2956 . . . . . . . . . . . . . . . . 17 ((∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐) → ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))
5436, 37, 53syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎))))) → ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))
55 fndm 5950 . . . . . . . . . . . . . . . . . 18 ( Fn 𝑐 → dom = 𝑐)
56 ineq12 3792 . . . . . . . . . . . . . . . . . . . 20 ((dom 𝑔 = 𝑏 ∧ dom = 𝑐) → (dom 𝑔 ∩ dom ) = (𝑏𝑐))
5756sseq1d 3616 . . . . . . . . . . . . . . . . . . 19 ((dom 𝑔 = 𝑏 ∧ dom = 𝑐) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ↔ (𝑏𝑐) ⊆ 𝐴))
58 sseq2 3611 . . . . . . . . . . . . . . . . . . . . 21 ((dom 𝑔 ∩ dom ) = (𝑏𝑐) → (Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ) ↔ Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))
5958raleqbi1dv 3140 . . . . . . . . . . . . . . . . . . . 20 ((dom 𝑔 ∩ dom ) = (𝑏𝑐) → (∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ) ↔ ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))
6056, 59syl 17 . . . . . . . . . . . . . . . . . . 19 ((dom 𝑔 = 𝑏 ∧ dom = 𝑐) → (∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ) ↔ ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)))
6157, 60anbi12d 746 . . . . . . . . . . . . . . . . . 18 ((dom 𝑔 = 𝑏 ∧ dom = 𝑐) → (((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )) ↔ ((𝑏𝑐) ⊆ 𝐴 ∧ ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))))
6212, 55, 61syl2an 494 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn 𝑏 Fn 𝑐) → (((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )) ↔ ((𝑏𝑐) ⊆ 𝐴 ∧ ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐))))
6362biimprcd 240 . . . . . . . . . . . . . . . 16 (((𝑏𝑐) ⊆ 𝐴 ∧ ∀𝑎 ∈ (𝑏𝑐)Pred(𝑅, 𝐴, 𝑎) ⊆ (𝑏𝑐)) → ((𝑔 Fn 𝑏 Fn 𝑐) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ))))
6435, 54, 63syl2anc 692 . . . . . . . . . . . . . . 15 (((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎))))) → ((𝑔 Fn 𝑏 Fn 𝑐) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom ))))
6564impcom 446 . . . . . . . . . . . . . 14 (((𝑔 Fn 𝑏 Fn 𝑐) ∧ ((𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))) ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎)))))) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
6665an4s 868 . . . . . . . . . . . . 13 (((𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))) ∧ ( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎)))))) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
6766exlimivv 1862 . . . . . . . . . . . 12 (∃𝑏𝑐((𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))) ∧ ( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎)))))) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
6832, 67sylbir 225 . . . . . . . . . . 11 ((∃𝑏(𝑔 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑎𝑏 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑏 ∧ ∀𝑎𝑏 (𝑔𝑎) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))) ∧ ∃𝑐( Fn 𝑐 ∧ (𝑐𝐴 ∧ ∀𝑎𝑐 Pred(𝑅, 𝐴, 𝑎) ⊆ 𝑐 ∧ ∀𝑎𝑐 (𝑎) = (𝑎𝐺( ↾ Pred(𝑅, 𝐴, 𝑎)))))) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
6911, 31, 68syl2anb 496 . . . . . . . . . 10 ((𝑔𝐵𝐵) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
7069adantr 481 . . . . . . . . 9 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((dom 𝑔 ∩ dom ) ⊆ 𝐴 ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )Pred(𝑅, 𝐴, 𝑎) ⊆ (dom 𝑔 ∩ dom )))
71 simpr 477 . . . . . . . . 9 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → 𝑎 ∈ (dom 𝑔 ∩ dom ))
72 preddowncl 5669 . . . . . . . . 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)))
7370, 71, 72sylc 65 . . . . . . . 8 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎) = Pred(𝑅, 𝐴, 𝑎))
7429, 73syl5eq 2672 . . . . . . 7 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((dom 𝑔 ∩ dom ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = Pred(𝑅, 𝐴, 𝑎))
7574reseq2d 5360 . . . . . 6 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → (𝑔 ↾ ((dom 𝑔 ∩ dom ) ∩ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))
7626, 75syl5eq 2672 . . . . 5 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑎)))
7776oveq2d 6621 . . . 4 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) = (𝑎𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑎))))
7823, 25, 773eqtr4d 2670 . . 3 (((𝑔𝐵𝐵) ∧ 𝑎 ∈ (dom 𝑔 ∩ dom )) → ((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))))
7978ralrimiva 2965 . 2 ((𝑔𝐵𝐵) → ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))))
807, 79jca 554 1 ((𝑔𝐵𝐵) → ((𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1992  {cab 2612  wral 2912  cin 3559  wss 3560  dom cdm 5079  cres 5081  Predcpred 5641  Fun wfun 5844   Fn wfn 5845  cfv 5850  (class class class)co 6605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-iota 5813  df-fun 5852  df-fn 5853  df-fv 5858  df-ov 6608
This theorem is referenced by:  frrlem5  31477
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