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Theorem ofoaf 43345
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 9684 . . . . 5 ω ∈ On
3 simpl 482 . . . . 5 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐷 ∈ On)
4 oecl 8574 . . . . 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 2839 . . 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 8623 . . . . . . . 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 2842 . . . . . 6 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∅ ∈ 𝐸)
12 oveq1 7438 . . . . . . . 8 (𝑥 = ∅ → (𝑥 +o 𝐸) = (∅ +o 𝐸))
1312sseq2d 4028 . . . . . . 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 8575 . . . . . . . 8 (𝐸 ∈ On → (∅ +o 𝐸) = 𝐸)
166, 15syl 17 . . . . . . 7 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (∅ +o 𝐸) = 𝐸)
17 ssid 4018 . . . . . . 7 (∅ +o 𝐸) ⊆ (∅ +o 𝐸)
1816, 17eqsstrrdi 4051 . . . . . 6 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 ⊆ (∅ +o 𝐸))
1911, 14, 18rspcedvd 3624 . . . . 5 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∃𝑥𝐸 𝐸 ⊆ (𝑥 +o 𝐸))
20 ssiun 5051 . . . . 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 2825 . . . . . . . 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 4018 . . . . . . . 8 𝐸𝐸
2725, 26eqsstrrdi 4051 . . . . . . 7 (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥𝐸) → (ω ↑o 𝐷) ⊆ 𝐸)
28 oaabs2 8686 . . . . . . 7 (((𝑥 ∈ (ω ↑o 𝐷) ∧ 𝐸 ∈ On) ∧ (ω ↑o 𝐷) ⊆ 𝐸) → (𝑥 +o 𝐸) = 𝐸)
2923, 24, 27, 28syl21anc 838 . . . . . 6 (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥𝐸) → (𝑥 +o 𝐸) = 𝐸)
3029, 26eqsstrdi 4050 . . . . 5 (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥𝐸) → (𝑥 +o 𝐸) ⊆ 𝐸)
3130iunssd 5055 . . . 4 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝑥𝐸 (𝑥 +o 𝐸) ⊆ 𝐸)
3221, 31eqssd 4013 . . 3 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 = 𝑥𝐸 (𝑥 +o 𝐸))
336, 6, 323jca 1127 . 2 ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (𝐸 ∈ On ∧ 𝐸 ∈ On ∧ 𝐸 = 𝑥𝐸 (𝑥 +o 𝐸)))
34 ofoafg 43344 . 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 1537  wcel 2106  wrex 3068  cin 3962  wss 3963  c0 4339   ciun 4996   × cxp 5687  cres 5691  Oncon0 6386  wf 6559  (class class class)co 7431  f cof 7695  ωcom 7887   +o coa 8502  o coe 8504  m cmap 8865
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-rmo 3378  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-int 4952  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-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511  df-map 8867
This theorem is referenced by:  ofoafo  43346  ofoacl  43347
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