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Theorem frrlem5 32121
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 3354 . . . . . 6 𝑥 ∈ V
2 vex 3354 . . . . . 6 𝑢 ∈ V
31, 2breldm 5467 . . . . 5 (𝑥𝑔𝑢𝑥 ∈ dom 𝑔)
4 vex 3354 . . . . . 6 𝑣 ∈ V
51, 4breldm 5467 . . . . 5 (𝑥𝑣𝑥 ∈ dom )
63, 5anim12i 600 . . . 4 ((𝑥𝑔𝑢𝑥𝑣) → (𝑥 ∈ dom 𝑔𝑥 ∈ dom ))
7 elin 3947 . . . 4 (𝑥 ∈ (dom 𝑔 ∩ dom ) ↔ (𝑥 ∈ dom 𝑔𝑥 ∈ dom ))
86, 7sylibr 224 . . 3 ((𝑥𝑔𝑢𝑥𝑣) → 𝑥 ∈ (dom 𝑔 ∩ dom ))
9 anandir 656 . . . 4 (((𝑥𝑔𝑢𝑥𝑣) ∧ 𝑥 ∈ (dom 𝑔 ∩ dom )) ↔ ((𝑥𝑔𝑢𝑥 ∈ (dom 𝑔 ∩ dom )) ∧ (𝑥𝑣𝑥 ∈ (dom 𝑔 ∩ dom ))))
102brres 5543 . . . . 5 (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢 ↔ (𝑥𝑔𝑢𝑥 ∈ (dom 𝑔 ∩ dom )))
114brres 5543 . . . . 5 (𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣 ↔ (𝑥𝑣𝑥 ∈ (dom 𝑔 ∩ dom )))
1210, 11anbi12i 612 . . . 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 667 . 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 32119 . . . . . . . 8 (𝑔𝐵 → dom 𝑔𝐴)
17 ssinss1 3990 . . . . . . . 8 (dom 𝑔𝐴 → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
18 frrlem5.1 . . . . . . . . . 10 𝑅 Fr 𝐴
19 frss 5216 . . . . . . . . . 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 5218 . . . . . . . . . 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 501 . . . . . . . 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 466 . . . . . 6 ((𝑔𝐵𝐵) → (𝑅 Fr (dom 𝑔 ∩ dom ) ∧ 𝑅 Se (dom 𝑔 ∩ dom )))
2715frrlem4 32120 . . . . . 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 32120 . . . . . . . 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 455 . . . . . . 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 3956 . . . . . . . . . . 11 (dom 𝑔 ∩ dom ) = (dom ∩ dom 𝑔)
3130reseq2i 5531 . . . . . . . . . 10 ( ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom ∩ dom 𝑔))
3231fneq1i 6125 . . . . . . . . 9 (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom 𝑔 ∩ dom ))
3330fneq2i 6126 . . . . . . . . 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 6333 . . . . . . . . . 10 (( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (( ↾ (dom ∩ dom 𝑔))‘𝑎)
36 predeq2 5826 . . . . . . . . . . . . 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 5532 . . . . . . . . . . 11 (( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = (( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))
3938oveq2i 6804 . . . . . . . . . 10 (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) = (𝑎𝐺(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))
4035, 39eqeq12i 2785 . . . . . . . . 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 3139 . . . . . . . 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 612 . . . . . . 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 32116 . . . . . 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 1476 . . . . 5 ((𝑔𝐵𝐵) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
4645breqd 4797 . . . 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 32118 . . . . 5 (𝑔𝐵 → Fun 𝑔)
49 funres 6072 . . . . 5 (Fun 𝑔 → Fun (𝑔 ↾ (dom 𝑔 ∩ dom )))
50 dffun2 6041 . . . . . 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 484 . . . . 5 (Fun (𝑔 ↾ (dom 𝑔 ∩ dom )) → ∀𝑥𝑢𝑣((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
52 2sp 2210 . . . . . 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 2209 . . . . 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 466 . . 3 ((𝑔𝐵𝐵) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
5647, 55sylan2d 592 . 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 382  w3a 1071  wal 1629   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wral 3061  cin 3722  wss 3723   class class class wbr 4786   Fr wfr 5205   Se wse 5206  dom cdm 5249  cres 5251  Rel wrel 5254  Predcpred 5822  Fun wfun 6025   Fn wfn 6026  cfv 6031  (class class class)co 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-trpred 32054
This theorem is referenced by:  frrlem5c  32123
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