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Theorem naddcnffo 43326
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 43324 . 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 7927 . . . . . . . . 9 ∅ ∈ ω
4 fconst6g 6810 . . . . . . . . 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 9455 . . . . . . . 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 2830 . . . . . . . . 9 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
11 eqid 2740 . . . . . . . . . 10 dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
12 omelon 9715 . . . . . . . . . . 11 ω ∈ On
1312a1i 11 . . . . . . . . . 10 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
1411, 13, 6cantnfs 9735 . . . . . . . . 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 712 . . . . . . 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 7617 . . . . . . . 8 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})) = (𝑓f +o (𝑋 × {∅})))
239eleq2d 2830 . . . . . . . . . . . . . . 15 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓𝑆𝑓 ∈ dom (ω CNF 𝑋)))
2411, 13, 6cantnfs 9735 . . . . . . . . . . . . . . 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 6748 . . . . . . . . . 10 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓 Fn 𝑋)
31 fnconstg 6809 . . . . . . . . . . 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 4248 . . . . . . . . . 10 (𝑋𝑋) = 𝑋
3530, 32, 33, 33, 34offn 7727 . . . . . . . . 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 774 . . . . . . . . . . 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 7731 . . . . . . . . . . 11 (((𝑓 Fn 𝑋 ∧ (𝑋 × {∅}) Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥𝑋)) → ((𝑓f +o (𝑋 × {∅}))‘𝑥) = ((𝑓𝑥) +o ((𝑋 × {∅})‘𝑥)))
4136, 37, 38, 39, 40syl22anc 838 . . . . . . . . . 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 7239 . . . . . . . . . . . 12 ((∅ ∈ ω ∧ 𝑥𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
4442, 39, 43syl2anc 583 . . . . . . . . . . 11 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
4544oveq2d 7464 . . . . . . . . . 10 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → ((𝑓𝑥) +o ((𝑋 × {∅})‘𝑥)) = ((𝑓𝑥) +o ∅))
4629ffvelcdmda 7118 . . . . . . . . . . 11 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → (𝑓𝑥) ∈ ω)
47 nnon 7909 . . . . . . . . . . 11 ((𝑓𝑥) ∈ ω → (𝑓𝑥) ∈ On)
48 oa0 8572 . . . . . . . . . . 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 2784 . . . . . . . . 9 ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥𝑋) → ((𝑓f +o (𝑋 × {∅}))‘𝑥) = (𝑓𝑥))
5135, 30, 50eqfnfvd 7067 . . . . . . . 8 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓f +o (𝑋 × {∅})) = 𝑓)
5222, 51eqtr2d 2781 . . . . . . 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 7456 . . . . . 6 (𝑧 = (𝑋 × {∅}) → (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧) = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})))
5655rspceeqv 3658 . . . . 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 7455 . . . . . . 7 (𝑔 = 𝑓 → (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
5958eqeq2d 2751 . . . . . 6 (𝑔 = 𝑓 → (𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) ↔ 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
6059rexbidv 3185 . . . . 5 (𝑔 = 𝑓 → (∃𝑧𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) ↔ ∃𝑧𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
6160rspcev 3635 . . . 4 ((𝑓𝑆 ∧ ∃𝑧𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)) → ∃𝑔𝑆𝑧𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
622, 57, 61syl2anc 583 . . 3 (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓𝑆) → ∃𝑔𝑆𝑧𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
6362ralrimiva 3152 . 2 ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∀𝑓𝑆𝑔𝑆𝑧𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
64 foov 7624 . 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 582 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 2108  wral 3067  wrex 3076  c0 4352  {csn 4648   class class class wbr 5166   × cxp 5698  dom cdm 5700  cres 5702  Oncon0 6395   Fn wfn 6568  wf 6569  ontowfo 6571  cfv 6573  (class class class)co 7448  f cof 7712  ωcom 7903   +o coa 8519   finSupp cfsupp 9431   CNF ccnf 9730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-seqom 8504  df-1o 8522  df-oadd 8526  df-map 8886  df-en 9004  df-fin 9007  df-fsupp 9432  df-cnf 9731
This theorem is referenced by: (None)
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