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Theorem wfrlem5 7625
Description: Lemma for well-founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfrlem5.1 𝑅 We 𝐴
wfrlem5.2 𝑅 Se 𝐴
wfrlem5.3 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
Assertion
Ref Expression
wfrlem5 ((𝑔𝐵𝐵) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Distinct variable groups:   𝐴,𝑓,𝑔,,𝑥,𝑦   𝑓,𝐹,𝑔,,𝑥,𝑦   𝑅,𝑓,𝑔,,𝑥,𝑦   𝑢,𝑔,𝑣,,𝑥
Allowed substitution hints:   𝐴(𝑣,𝑢)   𝐵(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔,)   𝑅(𝑣,𝑢)   𝐹(𝑣,𝑢)

Proof of Theorem wfrlem5
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 vex 3353 . . . . . 6 𝑥 ∈ V
2 vex 3353 . . . . . 6 𝑢 ∈ V
31, 2breldm 5499 . . . . 5 (𝑥𝑔𝑢𝑥 ∈ dom 𝑔)
4 vex 3353 . . . . . 6 𝑣 ∈ V
51, 4breldm 5499 . . . . 5 (𝑥𝑣𝑥 ∈ dom )
63, 5anim12i 606 . . . 4 ((𝑥𝑔𝑢𝑥𝑣) → (𝑥 ∈ dom 𝑔𝑥 ∈ dom ))
7 elin 3960 . . . 4 (𝑥 ∈ (dom 𝑔 ∩ dom ) ↔ (𝑥 ∈ dom 𝑔𝑥 ∈ dom ))
86, 7sylibr 225 . . 3 ((𝑥𝑔𝑢𝑥𝑣) → 𝑥 ∈ (dom 𝑔 ∩ dom ))
9 anandi 666 . . . 4 ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ (𝑥𝑔𝑢𝑥𝑣)) ↔ ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢) ∧ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣)))
102brresi 5576 . . . . 5 (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢 ↔ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢))
114brresi 5576 . . . . 5 (𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣 ↔ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣))
1210, 11anbi12i 620 . . . 4 ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣) ↔ ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢) ∧ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣)))
139, 12sylbb2 229 . . 3 ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣))
148, 13mpancom 679 . 2 ((𝑥𝑔𝑢𝑥𝑣) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣))
15 wfrlem5.3 . . . . . . . . 9 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
1615wfrlem3 7621 . . . . . . . 8 (𝑔𝐵 → dom 𝑔𝐴)
17 ssinss1 4003 . . . . . . . 8 (dom 𝑔𝐴 → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
18 wfrlem5.1 . . . . . . . . . 10 𝑅 We 𝐴
19 wess 5266 . . . . . . . . . 10 ((dom 𝑔 ∩ dom ) ⊆ 𝐴 → (𝑅 We 𝐴𝑅 We (dom 𝑔 ∩ dom )))
2018, 19mpi 20 . . . . . . . . 9 ((dom 𝑔 ∩ dom ) ⊆ 𝐴𝑅 We (dom 𝑔 ∩ dom ))
21 wfrlem5.2 . . . . . . . . . 10 𝑅 Se 𝐴
22 sess2 5248 . . . . . . . . . 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 507 . . . . . . . 8 ((dom 𝑔 ∩ dom ) ⊆ 𝐴 → (𝑅 We (dom 𝑔 ∩ dom ) ∧ 𝑅 Se (dom 𝑔 ∩ dom )))
2516, 17, 243syl 18 . . . . . . 7 (𝑔𝐵 → (𝑅 We (dom 𝑔 ∩ dom ) ∧ 𝑅 Se (dom 𝑔 ∩ dom )))
2625adantr 472 . . . . . 6 ((𝑔𝐵𝐵) → (𝑅 We (dom 𝑔 ∩ dom ) ∧ 𝑅 Se (dom 𝑔 ∩ dom )))
2715wfrlem4 7623 . . . . . 6 ((𝑔𝐵𝐵) → ((𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))
2815wfrlem4 7623 . . . . . . . 8 ((𝐵𝑔𝐵) → (( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔) ∧ ∀𝑎 ∈ (dom ∩ dom 𝑔)(( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝐹‘(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))))
2928ancoms 450 . . . . . . 7 ((𝑔𝐵𝐵) → (( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔) ∧ ∀𝑎 ∈ (dom ∩ dom 𝑔)(( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝐹‘(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))))
30 incom 3969 . . . . . . . . . . 11 (dom 𝑔 ∩ dom ) = (dom ∩ dom 𝑔)
3130reseq2i 5564 . . . . . . . . . 10 ( ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom ∩ dom 𝑔))
3231fneq1i 6165 . . . . . . . . 9 (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom 𝑔 ∩ dom ))
3330fneq2i 6166 . . . . . . . . 9 (( ↾ (dom ∩ dom 𝑔)) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔))
3432, 33bitri 266 . . . . . . . 8 (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔))
3531fveq1i 6378 . . . . . . . . . 10 (( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (( ↾ (dom ∩ dom 𝑔))‘𝑎)
36 predeq2 5870 . . . . . . . . . . . . 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 5565 . . . . . . . . . . 11 (( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = (( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))
3938fveq2i 6380 . . . . . . . . . 10 (𝐹‘(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) = (𝐹‘(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))
4035, 39eqeq12i 2779 . . . . . . . . 9 ((( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝐹‘(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) ↔ (( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝐹‘(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))))
4130, 40raleqbii 3137 . . . . . . . 8 (∀𝑎 ∈ (dom 𝑔 ∩ dom )(( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝐹‘(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) ↔ ∀𝑎 ∈ (dom ∩ dom 𝑔)(( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝐹‘(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))))
4234, 41anbi12i 620 . . . . . . 7 ((( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )(( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝐹‘(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))) ↔ (( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔) ∧ ∀𝑎 ∈ (dom ∩ dom 𝑔)(( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝐹‘(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))))
4329, 42sylibr 225 . . . . . 6 ((𝑔𝐵𝐵) → (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )(( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝐹‘(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))
44 wfr3g 7618 . . . . . 6 (((𝑅 We (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) = (𝐹‘((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) = (𝐹‘(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))))) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
4526, 27, 43, 44syl3anc 1490 . . . . 5 ((𝑔𝐵𝐵) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
4645breqd 4822 . . . 4 ((𝑔𝐵𝐵) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣))
4746biimprd 239 . . 3 ((𝑔𝐵𝐵) → (𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣))
4815wfrlem2 7620 . . . . 5 (𝑔𝐵 → Fun 𝑔)
49 funres 6112 . . . . 5 (Fun 𝑔 → Fun (𝑔 ↾ (dom 𝑔 ∩ dom )))
50 dffun2 6080 . . . . . 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 490 . . . . 5 (Fun (𝑔 ↾ (dom 𝑔 ∩ dom )) → ∀𝑥𝑢𝑣((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
52 2sp 2218 . . . . . 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 2217 . . . . 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 472 . . 3 ((𝑔𝐵𝐵) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
5647, 55sylan2d 598 . 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 1107  wal 1650   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wral 3055  cin 3733  wss 3734   class class class wbr 4811   Se wse 5236   We wwe 5237  dom cdm 5279  cres 5281  Rel wrel 5284  Predcpred 5866  Fun wfun 6064   Fn wfn 6065  cfv 6070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-iota 6033  df-fun 6072  df-fn 6073  df-fv 6078
This theorem is referenced by:  wfrfun  7631
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