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Theorem naddcnffo 43354
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 43352 . 2 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)
2 simpr 484 . . . 4 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓𝑆) → 𝑓𝑆)
3 peano1 7911 . . . . . . . . 9 ∅ ∈ ω
4 fconst6g 6798 . . . . . . . . 9 (∅ ∈ ω → (𝑋 × {∅}):𝑋⟶ω)
53, 4mp1i 13 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}):𝑋⟶ω)
6 simpl 482 . . . . . . . . 9 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
73a1i 11 . . . . . . . . 9 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∅ ∈ ω)
86, 7fczfsuppd 9424 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) finSupp ∅)
9 simpr 484 . . . . . . . . . 10 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
109eleq2d 2825 . . . . . . . . 9 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
11 eqid 2735 . . . . . . . . . 10 dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
12 omelon 9684 . . . . . . . . . . 11 ω ∈ On
1312a1i 11 . . . . . . . . . 10 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
1411, 13, 6cantnfs 9704 . . . . . . . . 9 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ dom (ω CNF 𝑋) ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
1510, 14bitrd 279 . . . . . . . 8 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
165, 8, 15mpbir2and 713 . . . . . . 7 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) ∈ 𝑆)
1716adantr 480 . . . . . 6 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓𝑆) → (𝑋 × {∅}) ∈ 𝑆)
18 simpl 482 . . . . . . . . . 10 ((𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝑓𝑆)
1918adantl 481 . . . . . . . . 9 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓𝑆)
20 simpr 484 . . . . . . . . . 10 ((𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝑋 × {∅}) ∈ 𝑆)
2120adantl 481 . . . . . . . . 9 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑋 × {∅}) ∈ 𝑆)
2219, 21ovresd 7600 . . . . . . . 8 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})) = (𝑓f +o (𝑋 × {∅})))
239eleq2d 2825 . . . . . . . . . . . . . . 15 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓𝑆𝑓 ∈ dom (ω CNF 𝑋)))
2411, 13, 6cantnfs 9704 . . . . . . . . . . . . . . 15 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ dom (ω CNF 𝑋) ↔ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
2523, 24bitrd 279 . . . . . . . . . . . . . 14 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓𝑆 ↔ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
2625biimpd 229 . . . . . . . . . . . . 13 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓𝑆 → (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
27 simpl 482 . . . . . . . . . . . . 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 406 . . . . . . . . . . 11 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓:𝑋⟶ω)
3029ffnd 6738 . . . . . . . . . 10 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓 Fn 𝑋)
31 fnconstg 6797 . . . . . . . . . . 11 (∅ ∈ ω → (𝑋 × {∅}) Fn 𝑋)
323, 31mp1i 13 . . . . . . . . . 10 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑋 × {∅}) Fn 𝑋)
336adantr 480 . . . . . . . . . 10 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑋 ∈ On)
34 inidm 4235 . . . . . . . . . 10 (𝑋𝑋) = 𝑋
3530, 32, 33, 33, 34offn 7710 . . . . . . . . 9 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓f +o (𝑋 × {∅})) Fn 𝑋)
3630adantr 480 . . . . . . . . . . 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 775 . . . . . . . . . . 11 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → 𝑋 ∈ On)
39 simpr 484 . . . . . . . . . . 11 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → 𝑥𝑋)
40 fnfvof 7714 . . . . . . . . . . 11 (((𝑓 Fn 𝑋 ∧ (𝑋 × {∅}) Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥𝑋)) → ((𝑓f +o (𝑋 × {∅}))‘𝑥) = ((𝑓𝑥) +o ((𝑋 × {∅})‘𝑥)))
4136, 37, 38, 39, 40syl22anc 839 . . . . . . . . . 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 7222 . . . . . . . . . . . 12 ((∅ ∈ ω ∧ 𝑥𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
4442, 39, 43syl2anc 584 . . . . . . . . . . 11 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
4544oveq2d 7447 . . . . . . . . . 10 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → ((𝑓𝑥) +o ((𝑋 × {∅})‘𝑥)) = ((𝑓𝑥) +o ∅))
4629ffvelcdmda 7104 . . . . . . . . . . 11 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → (𝑓𝑥) ∈ ω)
47 nnon 7893 . . . . . . . . . . 11 ((𝑓𝑥) ∈ ω → (𝑓𝑥) ∈ On)
48 oa0 8553 . . . . . . . . . . 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 2779 . . . . . . . . 9 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → ((𝑓f +o (𝑋 × {∅}))‘𝑥) = (𝑓𝑥))
5135, 30, 50eqfnfvd 7054 . . . . . . . 8 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓f +o (𝑋 × {∅})) = 𝑓)
5222, 51eqtr2d 2776 . . . . . . 7 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})))
5352expr 456 . . . . . 6 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓𝑆) → ((𝑋 × {∅}) ∈ 𝑆𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅}))))
5417, 53jcai 516 . . . . 5 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓𝑆) → ((𝑋 × {∅}) ∈ 𝑆𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅}))))
55 oveq2 7439 . . . . . 6 (𝑧 = (𝑋 × {∅}) → (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧) = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})))
5655rspceeqv 3645 . . . . 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 7438 . . . . . . 7 (𝑔 = 𝑓 → (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
5958eqeq2d 2746 . . . . . 6 (𝑔 = 𝑓 → (𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) ↔ 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
6059rexbidv 3177 . . . . 5 (𝑔 = 𝑓 → (∃𝑧𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) ↔ ∃𝑧𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
6160rspcev 3622 . . . 4 ((𝑓𝑆 ∧ ∃𝑧𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)) → ∃𝑔𝑆𝑧𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
622, 57, 61syl2anc 584 . . 3 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓𝑆) → ∃𝑔𝑆𝑧𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
6362ralrimiva 3144 . 2 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∀𝑓𝑆𝑔𝑆𝑧𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
64 foov 7607 . 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 583 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 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  c0 4339  {csn 4631   class class class wbr 5148   × cxp 5687  dom cdm 5689  cres 5691  Oncon0 6386   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431  f cof 7695  ωcom 7887   +o coa 8502   finSupp cfsupp 9399   CNF ccnf 9699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-oadd 8509  df-map 8867  df-en 8985  df-fin 8988  df-fsupp 9400  df-cnf 9700
This theorem is referenced by: (None)
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