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Theorem wfrlem5OLD 8316
Description: Lemma for well-ordered recursion. The values of two acceptable functions agree within their domains. Obsolete as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfrlem5OLD.1 𝑅 We 𝐴
wfrlem5OLD.2 𝑅 Se 𝐴
wfrlem5OLD.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
wfrlem5OLD ((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 wfrlem5OLD
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 vex 3477 . . . . . 6 𝑥 ∈ V
2 vex 3477 . . . . . 6 𝑢 ∈ V
31, 2breldm 5909 . . . . 5 (𝑥𝑔𝑢𝑥 ∈ dom 𝑔)
4 vex 3477 . . . . . 6 𝑣 ∈ V
51, 4breldm 5909 . . . . 5 (𝑥𝑣𝑥 ∈ dom )
63, 5anim12i 612 . . . 4 ((𝑥𝑔𝑢𝑥𝑣) → (𝑥 ∈ dom 𝑔𝑥 ∈ dom ))
7 elin 3965 . . . 4 (𝑥 ∈ (dom 𝑔 ∩ dom ) ↔ (𝑥 ∈ dom 𝑔𝑥 ∈ dom ))
86, 7sylibr 233 . . 3 ((𝑥𝑔𝑢𝑥𝑣) → 𝑥 ∈ (dom 𝑔 ∩ dom ))
9 anandi 673 . . . 4 ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ (𝑥𝑔𝑢𝑥𝑣)) ↔ ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢) ∧ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣)))
102brresi 5991 . . . . 5 (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢 ↔ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢))
114brresi 5991 . . . . 5 (𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣 ↔ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣))
1210, 11anbi12i 626 . . . 4 ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣) ↔ ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢) ∧ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣)))
139, 12sylbb2 237 . . 3 ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣))
148, 13mpancom 685 . 2 ((𝑥𝑔𝑢𝑥𝑣) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣))
15 wfrlem5OLD.3 . . . . . . . . 9 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
1615wfrlem3OLD 8313 . . . . . . . 8 (𝑔𝐵 → dom 𝑔𝐴)
17 ssinss1 4238 . . . . . . . 8 (dom 𝑔𝐴 → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
18 wfrlem5OLD.1 . . . . . . . . . 10 𝑅 We 𝐴
19 wess 5664 . . . . . . . . . 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 wfrlem5OLD.2 . . . . . . . . . 10 𝑅 Se 𝐴
22 sess2 5646 . . . . . . . . . 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 511 . . . . . . . 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 480 . . . . . 6 ((𝑔𝐵𝐵) → (𝑅 We (dom 𝑔 ∩ dom ) ∧ 𝑅 Se (dom 𝑔 ∩ dom )))
2715wfrlem4OLD 8315 . . . . . 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)))))
2815wfrlem4OLD 8315 . . . . . . . 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 458 . . . . . . 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 4202 . . . . . . . . . . 11 (dom 𝑔 ∩ dom ) = (dom ∩ dom 𝑔)
3130reseq2i 5979 . . . . . . . . . 10 ( ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom ∩ dom 𝑔))
3231fneq1i 6647 . . . . . . . . 9 (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom 𝑔 ∩ dom ))
3330fneq2i 6648 . . . . . . . . 9 (( ↾ (dom ∩ dom 𝑔)) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔))
3432, 33bitri 274 . . . . . . . 8 (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔))
3531fveq1i 6893 . . . . . . . . . 10 (( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (( ↾ (dom ∩ dom 𝑔))‘𝑎)
36 predeq2 6304 . . . . . . . . . . . . 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 5980 . . . . . . . . . . 11 (( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = (( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))
3938fveq2i 6895 . . . . . . . . . 10 (𝐹‘(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) = (𝐹‘(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))
4035, 39eqeq12i 2749 . . . . . . . . 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 3337 . . . . . . . 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 626 . . . . . . 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 233 . . . . . 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 8310 . . . . . 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 1370 . . . . 5 ((𝑔𝐵𝐵) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
4645breqd 5160 . . . 4 ((𝑔𝐵𝐵) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣))
4746biimprd 247 . . 3 ((𝑔𝐵𝐵) → (𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣))
4815wfrlem2OLD 8312 . . . . 5 (𝑔𝐵 → Fun 𝑔)
49 funres 6591 . . . . 5 (Fun 𝑔 → Fun (𝑔 ↾ (dom 𝑔 ∩ dom )))
50 dffun2 6554 . . . . . 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 496 . . . . 5 (Fun (𝑔 ↾ (dom 𝑔 ∩ dom )) → ∀𝑥𝑢𝑣((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
52 2sp 2178 . . . . . 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 2177 . . . . 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 480 . . 3 ((𝑔𝐵𝐵) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
5647, 55sylan2d 604 . 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 395  w3a 1086  wal 1538   = wceq 1540  wex 1780  wcel 2105  {cab 2708  wral 3060  cin 3948  wss 3949   class class class wbr 5149   Se wse 5630   We wwe 5631  dom cdm 5677  cres 5679  Rel wrel 5682  Predcpred 6300  Fun wfun 6538   Fn wfn 6539  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552
This theorem is referenced by:  wfrfunOLD  8322
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