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Theorem frrlem5 31758
Description: Lemma for founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Paul Chapman, 21-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
frrlem5.1 𝑅 Fr 𝐴
frrlem5.2 𝑅 Se 𝐴
frrlem5.3 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
Assertion
Ref Expression
frrlem5 ((𝑔𝐵𝐵) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Distinct variable groups:   𝐴,𝑓,𝑔,,𝑥,𝑦   𝑓,𝐺,,𝑥,𝑦,𝑔   𝑢,𝑔,𝑣,𝑥   𝑦,𝑔   𝑢,,𝑣   𝑅,𝑓,𝑔,,𝑥,𝑦   𝐵,𝑔,,𝑢,𝑣,𝑥
Allowed substitution hints:   𝐴(𝑣,𝑢)   𝐵(𝑦,𝑓)   𝑅(𝑣,𝑢)   𝐺(𝑣,𝑢)

Proof of Theorem frrlem5
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 vex 3198 . . . . . 6 𝑥 ∈ V
2 vex 3198 . . . . . 6 𝑢 ∈ V
31, 2breldm 5318 . . . . 5 (𝑥𝑔𝑢𝑥 ∈ dom 𝑔)
4 vex 3198 . . . . . 6 𝑣 ∈ V
51, 4breldm 5318 . . . . 5 (𝑥𝑣𝑥 ∈ dom )
63, 5anim12i 589 . . . 4 ((𝑥𝑔𝑢𝑥𝑣) → (𝑥 ∈ dom 𝑔𝑥 ∈ dom ))
7 elin 3788 . . . 4 (𝑥 ∈ (dom 𝑔 ∩ dom ) ↔ (𝑥 ∈ dom 𝑔𝑥 ∈ dom ))
86, 7sylibr 224 . . 3 ((𝑥𝑔𝑢𝑥𝑣) → 𝑥 ∈ (dom 𝑔 ∩ dom ))
9 anandir 871 . . . 4 (((𝑥𝑔𝑢𝑥𝑣) ∧ 𝑥 ∈ (dom 𝑔 ∩ dom )) ↔ ((𝑥𝑔𝑢𝑥 ∈ (dom 𝑔 ∩ dom )) ∧ (𝑥𝑣𝑥 ∈ (dom 𝑔 ∩ dom ))))
102brres 5391 . . . . 5 (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢 ↔ (𝑥𝑔𝑢𝑥 ∈ (dom 𝑔 ∩ dom )))
114brres 5391 . . . . 5 (𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣 ↔ (𝑥𝑣𝑥 ∈ (dom 𝑔 ∩ dom )))
1210, 11anbi12i 732 . . . 4 ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣) ↔ ((𝑥𝑔𝑢𝑥 ∈ (dom 𝑔 ∩ dom )) ∧ (𝑥𝑣𝑥 ∈ (dom 𝑔 ∩ dom ))))
139, 12sylbb2 228 . . 3 (((𝑥𝑔𝑢𝑥𝑣) ∧ 𝑥 ∈ (dom 𝑔 ∩ dom )) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣))
148, 13mpdan 701 . 2 ((𝑥𝑔𝑢𝑥𝑣) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣))
15 frrlem5.3 . . . . . . . . 9 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
1615frrlem3 31756 . . . . . . . 8 (𝑔𝐵 → dom 𝑔𝐴)
17 ssinss1 3833 . . . . . . . 8 (dom 𝑔𝐴 → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
18 frrlem5.1 . . . . . . . . . 10 𝑅 Fr 𝐴
19 frss 5071 . . . . . . . . . 10 ((dom 𝑔 ∩ dom ) ⊆ 𝐴 → (𝑅 Fr 𝐴𝑅 Fr (dom 𝑔 ∩ dom )))
2018, 19mpi 20 . . . . . . . . 9 ((dom 𝑔 ∩ dom ) ⊆ 𝐴𝑅 Fr (dom 𝑔 ∩ dom ))
21 frrlem5.2 . . . . . . . . . 10 𝑅 Se 𝐴
22 sess2 5073 . . . . . . . . . 10 ((dom 𝑔 ∩ dom ) ⊆ 𝐴 → (𝑅 Se 𝐴𝑅 Se (dom 𝑔 ∩ dom )))
2321, 22mpi 20 . . . . . . . . 9 ((dom 𝑔 ∩ dom ) ⊆ 𝐴𝑅 Se (dom 𝑔 ∩ dom ))
2420, 23jca 554 . . . . . . . 8 ((dom 𝑔 ∩ dom ) ⊆ 𝐴 → (𝑅 Fr (dom 𝑔 ∩ dom ) ∧ 𝑅 Se (dom 𝑔 ∩ dom )))
2516, 17, 243syl 18 . . . . . . 7 (𝑔𝐵 → (𝑅 Fr (dom 𝑔 ∩ dom ) ∧ 𝑅 Se (dom 𝑔 ∩ dom )))
2625adantr 481 . . . . . 6 ((𝑔𝐵𝐵) → (𝑅 Fr (dom 𝑔 ∩ dom ) ∧ 𝑅 Se (dom 𝑔 ∩ dom )))
2715frrlem4 31757 . . . . . 6 ((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)))))
2815frrlem4 31757 . . . . . . . 8 ((𝐵𝑔𝐵) → (( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔) ∧ ∀𝑎 ∈ (dom ∩ dom 𝑔)(( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝑎𝐺(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))))
2928ancoms 469 . . . . . . 7 ((𝑔𝐵𝐵) → (( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔) ∧ ∀𝑎 ∈ (dom ∩ dom 𝑔)(( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝑎𝐺(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))))
30 incom 3797 . . . . . . . . . . 11 (dom 𝑔 ∩ dom ) = (dom ∩ dom 𝑔)
3130reseq2i 5382 . . . . . . . . . 10 ( ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom ∩ dom 𝑔))
3231fneq1i 5973 . . . . . . . . 9 (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom 𝑔 ∩ dom ))
3330fneq2i 5974 . . . . . . . . 9 (( ↾ (dom ∩ dom 𝑔)) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔))
3432, 33bitri 264 . . . . . . . 8 (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔))
3531fveq1i 6179 . . . . . . . . . 10 (( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (( ↾ (dom ∩ dom 𝑔))‘𝑎)
36 predeq2 5671 . . . . . . . . . . . . 13 ((dom 𝑔 ∩ dom ) = (dom ∩ dom 𝑔) → Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎) = Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))
3730, 36ax-mp 5 . . . . . . . . . . . 12 Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎) = Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)
3831, 37reseq12i 5383 . . . . . . . . . . 11 (( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = (( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))
3938oveq2i 6646 . . . . . . . . . 10 (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) = (𝑎𝐺(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))
4035, 39eqeq12i 2634 . . . . . . . . 9 ((( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) ↔ (( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝑎𝐺(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))))
4130, 40raleqbii 2987 . . . . . . . 8 (∀𝑎 ∈ (dom 𝑔 ∩ dom )(( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) ↔ ∀𝑎 ∈ (dom ∩ dom 𝑔)(( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝑎𝐺(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))))
4234, 41anbi12i 732 . . . . . . 7 ((( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )(( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))) ↔ (( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔) ∧ ∀𝑎 ∈ (dom ∩ dom 𝑔)(( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝑎𝐺(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))))
4329, 42sylibr 224 . . . . . 6 ((𝑔𝐵𝐵) → (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )(( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))
44 frr3g 31753 . . . . . 6 (((𝑅 Fr (dom 𝑔 ∩ dom ) ∧ 𝑅 Se (dom 𝑔 ∩ dom )) ∧ ((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)))) ∧ (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )(( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))))) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
4526, 27, 43, 44syl3anc 1324 . . . . 5 ((𝑔𝐵𝐵) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
4645breqd 4655 . . . 4 ((𝑔𝐵𝐵) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣))
4746biimprd 238 . . 3 ((𝑔𝐵𝐵) → (𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣))
4815frrlem2 31755 . . . . 5 (𝑔𝐵 → Fun 𝑔)
49 funres 5917 . . . . 5 (Fun 𝑔 → Fun (𝑔 ↾ (dom 𝑔 ∩ dom )))
50 dffun2 5886 . . . . . 6 (Fun (𝑔 ↾ (dom 𝑔 ∩ dom )) ↔ (Rel (𝑔 ↾ (dom 𝑔 ∩ dom )) ∧ ∀𝑥𝑢𝑣((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣)))
5150simprbi 480 . . . . 5 (Fun (𝑔 ↾ (dom 𝑔 ∩ dom )) → ∀𝑥𝑢𝑣((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
52 2sp 2054 . . . . . 6 (∀𝑢𝑣((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
5352sps 2053 . . . . 5 (∀𝑥𝑢𝑣((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
5448, 49, 51, 534syl 19 . . . 4 (𝑔𝐵 → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
5554adantr 481 . . 3 ((𝑔𝐵𝐵) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
5647, 55sylan2d 499 . 2 ((𝑔𝐵𝐵) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
5714, 56syl5 34 1 ((𝑔𝐵𝐵) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wal 1479   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wral 2909  cin 3566  wss 3567   class class class wbr 4644   Fr wfr 5060   Se wse 5061  dom cdm 5104  cres 5106  Rel wrel 5109  Predcpred 5667  Fun wfun 5870   Fn wfn 5871  cfv 5876  (class class class)co 6635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-trpred 31692
This theorem is referenced by:  frrlem5c  31760
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