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Theorem naddcnffo 42858
Description: Addition of Cantor normal forms is a function onto Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
Assertion
Ref Expression
naddcnffo ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)–onto𝑆)

Proof of Theorem naddcnffo
Dummy variables 𝑓 𝑔 𝑧 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 naddcnff 42856 . 2 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)
2 simpr 483 . . . 4 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓𝑆) → 𝑓𝑆)
3 peano1 7892 . . . . . . . . 9 ∅ ∈ ω
4 fconst6g 6781 . . . . . . . . 9 (∅ ∈ ω → (𝑋 × {∅}):𝑋⟶ω)
53, 4mp1i 13 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}):𝑋⟶ω)
6 simpl 481 . . . . . . . . 9 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
73a1i 11 . . . . . . . . 9 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∅ ∈ ω)
86, 7fczfsuppd 9409 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) finSupp ∅)
9 simpr 483 . . . . . . . . . 10 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
109eleq2d 2811 . . . . . . . . 9 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
11 eqid 2725 . . . . . . . . . 10 dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
12 omelon 9669 . . . . . . . . . . 11 ω ∈ On
1312a1i 11 . . . . . . . . . 10 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
1411, 13, 6cantnfs 9689 . . . . . . . . 9 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ dom (ω CNF 𝑋) ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
1510, 14bitrd 278 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
165, 8, 15mpbir2and 711 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) ∈ 𝑆)
1716adantr 479 . . . . . 6 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓𝑆) → (𝑋 × {∅}) ∈ 𝑆)
18 simpl 481 . . . . . . . . . 10 ((𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝑓𝑆)
1918adantl 480 . . . . . . . . 9 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓𝑆)
20 simpr 483 . . . . . . . . . 10 ((𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝑋 × {∅}) ∈ 𝑆)
2120adantl 480 . . . . . . . . 9 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑋 × {∅}) ∈ 𝑆)
2219, 21ovresd 7585 . . . . . . . 8 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})) = (𝑓f +o (𝑋 × {∅})))
239eleq2d 2811 . . . . . . . . . . . . . . 15 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓𝑆𝑓 ∈ dom (ω CNF 𝑋)))
2411, 13, 6cantnfs 9689 . . . . . . . . . . . . . . 15 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ dom (ω CNF 𝑋) ↔ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
2523, 24bitrd 278 . . . . . . . . . . . . . 14 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓𝑆 ↔ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
2625biimpd 228 . . . . . . . . . . . . 13 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓𝑆 → (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
27 simpl 481 . . . . . . . . . . . . 13 ((𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅) → 𝑓:𝑋⟶ω)
2818, 26, 27syl56 36 . . . . . . . . . . . 12 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝑓:𝑋⟶ω))
2928imp 405 . . . . . . . . . . 11 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓:𝑋⟶ω)
3029ffnd 6718 . . . . . . . . . 10 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓 Fn 𝑋)
31 fnconstg 6780 . . . . . . . . . . 11 (∅ ∈ ω → (𝑋 × {∅}) Fn 𝑋)
323, 31mp1i 13 . . . . . . . . . 10 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑋 × {∅}) Fn 𝑋)
336adantr 479 . . . . . . . . . 10 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑋 ∈ On)
34 inidm 4213 . . . . . . . . . 10 (𝑋𝑋) = 𝑋
3530, 32, 33, 33, 34offn 7695 . . . . . . . . 9 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓f +o (𝑋 × {∅})) Fn 𝑋)
3630adantr 479 . . . . . . . . . . 11 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → 𝑓 Fn 𝑋)
373, 31mp1i 13 . . . . . . . . . . 11 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → (𝑋 × {∅}) Fn 𝑋)
38 simplll 773 . . . . . . . . . . 11 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → 𝑋 ∈ On)
39 simpr 483 . . . . . . . . . . 11 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → 𝑥𝑋)
40 fnfvof 7699 . . . . . . . . . . 11 (((𝑓 Fn 𝑋 ∧ (𝑋 × {∅}) Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥𝑋)) → ((𝑓f +o (𝑋 × {∅}))‘𝑥) = ((𝑓𝑥) +o ((𝑋 × {∅})‘𝑥)))
4136, 37, 38, 39, 40syl22anc 837 . . . . . . . . . 10 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → ((𝑓f +o (𝑋 × {∅}))‘𝑥) = ((𝑓𝑥) +o ((𝑋 × {∅})‘𝑥)))
423a1i 11 . . . . . . . . . . . 12 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → ∅ ∈ ω)
43 fvconst2g 7210 . . . . . . . . . . . 12 ((∅ ∈ ω ∧ 𝑥𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
4442, 39, 43syl2anc 582 . . . . . . . . . . 11 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
4544oveq2d 7432 . . . . . . . . . 10 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → ((𝑓𝑥) +o ((𝑋 × {∅})‘𝑥)) = ((𝑓𝑥) +o ∅))
4629ffvelcdmda 7089 . . . . . . . . . . 11 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → (𝑓𝑥) ∈ ω)
47 nnon 7874 . . . . . . . . . . 11 ((𝑓𝑥) ∈ ω → (𝑓𝑥) ∈ On)
48 oa0 8535 . . . . . . . . . . 11 ((𝑓𝑥) ∈ On → ((𝑓𝑥) +o ∅) = (𝑓𝑥))
4946, 47, 483syl 18 . . . . . . . . . 10 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → ((𝑓𝑥) +o ∅) = (𝑓𝑥))
5041, 45, 493eqtrd 2769 . . . . . . . . 9 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → ((𝑓f +o (𝑋 × {∅}))‘𝑥) = (𝑓𝑥))
5135, 30, 50eqfnfvd 7038 . . . . . . . 8 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓f +o (𝑋 × {∅})) = 𝑓)
5222, 51eqtr2d 2766 . . . . . . 7 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})))
5352expr 455 . . . . . 6 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓𝑆) → ((𝑋 × {∅}) ∈ 𝑆𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅}))))
5417, 53jcai 515 . . . . 5 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓𝑆) → ((𝑋 × {∅}) ∈ 𝑆𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅}))))
55 oveq2 7424 . . . . . 6 (𝑧 = (𝑋 × {∅}) → (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧) = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})))
5655rspceeqv 3623 . . . . 5 (((𝑋 × {∅}) ∈ 𝑆𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅}))) → ∃𝑧𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
5754, 56syl 17 . . . 4 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓𝑆) → ∃𝑧𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
58 oveq1 7423 . . . . . . 7 (𝑔 = 𝑓 → (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
5958eqeq2d 2736 . . . . . 6 (𝑔 = 𝑓 → (𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) ↔ 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
6059rexbidv 3169 . . . . 5 (𝑔 = 𝑓 → (∃𝑧𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) ↔ ∃𝑧𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
6160rspcev 3601 . . . 4 ((𝑓𝑆 ∧ ∃𝑧𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)) → ∃𝑔𝑆𝑧𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
622, 57, 61syl2anc 582 . . 3 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓𝑆) → ∃𝑔𝑆𝑧𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
6362ralrimiva 3136 . 2 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∀𝑓𝑆𝑔𝑆𝑧𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
64 foov 7592 . 2 (( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)–onto𝑆 ↔ (( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆 ∧ ∀𝑓𝑆𝑔𝑆𝑧𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
651, 63, 64sylanbrc 581 1 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)–onto𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3051  wrex 3060  c0 4318  {csn 4624   class class class wbr 5143   × cxp 5670  dom cdm 5672  cres 5674  Oncon0 6364   Fn wfn 6538  wf 6539  ontowfo 6541  cfv 6543  (class class class)co 7416  f cof 7680  ωcom 7868   +o coa 8482   finSupp cfsupp 9385   CNF ccnf 9684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-om 7869  df-1st 7991  df-2nd 7992  df-supp 8164  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-seqom 8467  df-1o 8485  df-oadd 8489  df-map 8845  df-en 8963  df-fin 8966  df-fsupp 9386  df-cnf 9685
This theorem is referenced by: (None)
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