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Theorem ofoaf 43373
Description: Addition operator for functions from sets into power of omega results in a function from the intersection of sets to that power of omega. (Contributed by RP, 5-Jan-2025.)
Assertion
Ref Expression
ofoaf (((𝐴𝑉𝐵𝑊𝐶 = (𝐴𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷))) → ( ∘f +o ↾ ((𝐸m 𝐴) × (𝐸m 𝐵))):((𝐸m 𝐴) × (𝐸m 𝐵))⟶(𝐸m 𝐶))

Proof of Theorem ofoaf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 484 . . . 4 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 = (ω ↑o 𝐷))
2 omelon 9687 . . . . 5 ω ∈ On
3 simpl 482 . . . . 5 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐷 ∈ On)
4 oecl 8576 . . . . 5 ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ↑o 𝐷) ∈ On)
52, 3, 4sylancr 587 . . . 4 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (ω ↑o 𝐷) ∈ On)
61, 5eqeltrd 2840 . . 3 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 ∈ On)
73, 2jctil 519 . . . . . . . 8 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (ω ∈ On ∧ 𝐷 ∈ On))
8 peano1 7911 . . . . . . . 8 ∅ ∈ ω
9 oen0 8625 . . . . . . . 8 (((ω ∈ On ∧ 𝐷 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐷))
107, 8, 9sylancl 586 . . . . . . 7 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∅ ∈ (ω ↑o 𝐷))
1110, 1eleqtrrd 2843 . . . . . 6 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∅ ∈ 𝐸)
12 oveq1 7439 . . . . . . . 8 (𝑥 = ∅ → (𝑥 +o 𝐸) = (∅ +o 𝐸))
1312sseq2d 4015 . . . . . . 7 (𝑥 = ∅ → (𝐸 ⊆ (𝑥 +o 𝐸) ↔ 𝐸 ⊆ (∅ +o 𝐸)))
1413adantl 481 . . . . . 6 (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 = ∅) → (𝐸 ⊆ (𝑥 +o 𝐸) ↔ 𝐸 ⊆ (∅ +o 𝐸)))
15 oa0r 8577 . . . . . . . 8 (𝐸 ∈ On → (∅ +o 𝐸) = 𝐸)
166, 15syl 17 . . . . . . 7 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (∅ +o 𝐸) = 𝐸)
17 ssid 4005 . . . . . . 7 (∅ +o 𝐸) ⊆ (∅ +o 𝐸)
1816, 17eqsstrrdi 4028 . . . . . 6 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 ⊆ (∅ +o 𝐸))
1911, 14, 18rspcedvd 3623 . . . . 5 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∃𝑥𝐸 𝐸 ⊆ (𝑥 +o 𝐸))
20 ssiun 5045 . . . . 5 (∃𝑥𝐸 𝐸 ⊆ (𝑥 +o 𝐸) → 𝐸 𝑥𝐸 (𝑥 +o 𝐸))
2119, 20syl 17 . . . 4 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 𝑥𝐸 (𝑥 +o 𝐸))
221eleq2d 2826 . . . . . . . 8 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (𝑥𝐸𝑥 ∈ (ω ↑o 𝐷)))
2322biimpa 476 . . . . . . 7 (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥𝐸) → 𝑥 ∈ (ω ↑o 𝐷))
246adantr 480 . . . . . . 7 (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥𝐸) → 𝐸 ∈ On)
251adantr 480 . . . . . . . 8 (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥𝐸) → 𝐸 = (ω ↑o 𝐷))
26 ssid 4005 . . . . . . . 8 𝐸𝐸
2725, 26eqsstrrdi 4028 . . . . . . 7 (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥𝐸) → (ω ↑o 𝐷) ⊆ 𝐸)
28 oaabs2 8688 . . . . . . 7 (((𝑥 ∈ (ω ↑o 𝐷) ∧ 𝐸 ∈ On) ∧ (ω ↑o 𝐷) ⊆ 𝐸) → (𝑥 +o 𝐸) = 𝐸)
2923, 24, 27, 28syl21anc 837 . . . . . 6 (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥𝐸) → (𝑥 +o 𝐸) = 𝐸)
3029, 26eqsstrdi 4027 . . . . 5 (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥𝐸) → (𝑥 +o 𝐸) ⊆ 𝐸)
3130iunssd 5049 . . . 4 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝑥𝐸 (𝑥 +o 𝐸) ⊆ 𝐸)
3221, 31eqssd 4000 . . 3 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 = 𝑥𝐸 (𝑥 +o 𝐸))
336, 6, 323jca 1128 . 2 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (𝐸 ∈ On ∧ 𝐸 ∈ On ∧ 𝐸 = 𝑥𝐸 (𝑥 +o 𝐸)))
34 ofoafg 43372 . 2 (((𝐴𝑉𝐵𝑊𝐶 = (𝐴𝐵)) ∧ (𝐸 ∈ On ∧ 𝐸 ∈ On ∧ 𝐸 = 𝑥𝐸 (𝑥 +o 𝐸))) → ( ∘f +o ↾ ((𝐸m 𝐴) × (𝐸m 𝐵))):((𝐸m 𝐴) × (𝐸m 𝐵))⟶(𝐸m 𝐶))
3533, 34sylan2 593 1 (((𝐴𝑉𝐵𝑊𝐶 = (𝐴𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷))) → ( ∘f +o ↾ ((𝐸m 𝐴) × (𝐸m 𝐵))):((𝐸m 𝐴) × (𝐸m 𝐵))⟶(𝐸m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wrex 3069  cin 3949  wss 3950  c0 4332   ciun 4990   × cxp 5682  cres 5686  Oncon0 6383  wf 6556  (class class class)co 7432  f cof 7696  ωcom 7888   +o coa 8504  o coe 8506  m cmap 8867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-omul 8512  df-oexp 8513  df-map 8869
This theorem is referenced by:  ofoafo  43374  ofoacl  43375
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