Theorem ofoaf 44110

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	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷))) → ( ∘f +o ↾ ((𝐸 ↑m 𝐴) × (𝐸 ↑m 𝐵))):((𝐸 ↑m 𝐴) × (𝐸 ↑m 𝐵))⟶(𝐸 ↑m 𝐶))

Proof of Theorem ofoaf

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 489	. . . 4 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 = (ω ↑o 𝐷))
2		omelon 9613	. . . . 5 ⊢ ω ∈ On
3		simpl 487	. . . . 5 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐷 ∈ On)
4		oecl 8520	. . . . 5 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ↑o 𝐷) ∈ On)
5	2, 3, 4	sylancr 598	. . . 4 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (ω ↑o 𝐷) ∈ On)
6	1, 5	eqeltrd 2862	. . 3 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 ∈ On)
7	3, 2	jctil 528	. . . . . . . 8 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (ω ∈ On ∧ 𝐷 ∈ On))
8		peano1 7883	. . . . . . . 8 ⊢ ∅ ∈ ω
9		oen0 8570	. . . . . . . 8 ⊢ (((ω ∈ On ∧ 𝐷 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐷))
10	7, 8, 9	sylancl 597	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∅ ∈ (ω ↑o 𝐷))
11	10, 1	eleqtrrd 2865	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∅ ∈ 𝐸)
12		oveq1 7419	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 +o 𝐸) = (∅ +o 𝐸))
13	12	sseq2d 3968	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐸 ⊆ (𝑥 +o 𝐸) ↔ 𝐸 ⊆ (∅ +o 𝐸)))
14	13	adantl 486	. . . . . 6 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 = ∅) → (𝐸 ⊆ (𝑥 +o 𝐸) ↔ 𝐸 ⊆ (∅ +o 𝐸)))
15		oa0r 8521	. . . . . . . 8 ⊢ (𝐸 ∈ On → (∅ +o 𝐸) = 𝐸)
16	6, 15	syl 18	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (∅ +o 𝐸) = 𝐸)
17		ssid 3958	. . . . . . 7 ⊢ (∅ +o 𝐸) ⊆ (∅ +o 𝐸)
18	16, 17	eqsstrrdi 3981	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 ⊆ (∅ +o 𝐸))
19	11, 14, 18	rspcedvd 3582	. . . . 5 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∃𝑥 ∈ 𝐸 𝐸 ⊆ (𝑥 +o 𝐸))
20		ssiun 5010	. . . . 5 ⊢ (∃𝑥 ∈ 𝐸 𝐸 ⊆ (𝑥 +o 𝐸) → 𝐸 ⊆ ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸))
21	19, 20	syl 18	. . . 4 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 ⊆ ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸))
22	1	eleq2d 2848	. . . . . . . 8 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (𝑥 ∈ 𝐸 ↔ 𝑥 ∈ (ω ↑o 𝐷)))
23	22	biimpa 481	. . . . . . 7 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → 𝑥 ∈ (ω ↑o 𝐷))
24	6	adantr 485	. . . . . . 7 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → 𝐸 ∈ On)
25	1	adantr 485	. . . . . . . 8 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → 𝐸 = (ω ↑o 𝐷))
26		ssid 3958	. . . . . . . 8 ⊢ 𝐸 ⊆ 𝐸
27	25, 26	eqsstrrdi 3981	. . . . . . 7 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → (ω ↑o 𝐷) ⊆ 𝐸)
28		oaabs2 8633	. . . . . . 7 ⊢ (((𝑥 ∈ (ω ↑o 𝐷) ∧ 𝐸 ∈ On) ∧ (ω ↑o 𝐷) ⊆ 𝐸) → (𝑥 +o 𝐸) = 𝐸)
29	23, 24, 27, 28	syl21anc 850	. . . . . 6 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → (𝑥 +o 𝐸) = 𝐸)
30	29, 26	eqsstrdi 3980	. . . . 5 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → (𝑥 +o 𝐸) ⊆ 𝐸)
31	30	iunssd 5014	. . . 4 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸) ⊆ 𝐸)
32	21, 31	eqssd 3953	. . 3 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 = ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸))
33	6, 6, 32	3jca 1145	. 2 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (𝐸 ∈ On ∧ 𝐸 ∈ On ∧ 𝐸 = ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸)))
34		ofoafg 44109	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐸 ∈ On ∧ 𝐸 ∈ On ∧ 𝐸 = ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸))) → ( ∘f +o ↾ ((𝐸 ↑m 𝐴) × (𝐸 ↑m 𝐵))):((𝐸 ↑m 𝐴) × (𝐸 ↑m 𝐵))⟶(𝐸 ↑m 𝐶))
35	33, 34	sylan2 604	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷))) → ( ∘f +o ↾ ((𝐸 ↑m 𝐴) × (𝐸 ↑m 𝐵))):((𝐸 ↑m 𝐴) × (𝐸 ↑m 𝐵))⟶(𝐸 ↑m 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∃wrex 3088   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ∪ ciun 4955   × cxp 5658   ↾ cres 5662  Oncon0 6360  ⟶wf 6532  (class class class)co 7412   ∘_f cof 7674  ωcom 7860   +_o coa 8448   ↑_o coe 8450   ↑_m cmap 8822

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-inf2 9608

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-oadd 8455  df-omul 8456  df-oexp 8457  df-map 8824

This theorem is used by:  ofoafo  44111  ofoacl  44112