Theorem ofoaf 42090

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 485	. . . 4 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 = (ω ↑o 𝐷))
2		omelon 9637	. . . . 5 ⊢ ω ∈ On
3		simpl 483	. . . . 5 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐷 ∈ On)
4		oecl 8533	. . . . 5 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ↑o 𝐷) ∈ On)
5	2, 3, 4	sylancr 587	. . . 4 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (ω ↑o 𝐷) ∈ On)
6	1, 5	eqeltrd 2833	. . 3 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 ∈ On)
7	3, 2	jctil 520	. . . . . . . 8 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (ω ∈ On ∧ 𝐷 ∈ On))
8		peano1 7875	. . . . . . . 8 ⊢ ∅ ∈ ω
9		oen0 8582	. . . . . . . 8 ⊢ (((ω ∈ On ∧ 𝐷 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐷))
10	7, 8, 9	sylancl 586	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∅ ∈ (ω ↑o 𝐷))
11	10, 1	eleqtrrd 2836	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∅ ∈ 𝐸)
12		oveq1 7412	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 +o 𝐸) = (∅ +o 𝐸))
13	12	sseq2d 4013	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐸 ⊆ (𝑥 +o 𝐸) ↔ 𝐸 ⊆ (∅ +o 𝐸)))
14	13	adantl 482	. . . . . 6 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 = ∅) → (𝐸 ⊆ (𝑥 +o 𝐸) ↔ 𝐸 ⊆ (∅ +o 𝐸)))
15		oa0r 8534	. . . . . . . 8 ⊢ (𝐸 ∈ On → (∅ +o 𝐸) = 𝐸)
16	6, 15	syl 17	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (∅ +o 𝐸) = 𝐸)
17		ssid 4003	. . . . . . 7 ⊢ (∅ +o 𝐸) ⊆ (∅ +o 𝐸)
18	16, 17	eqsstrrdi 4036	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 ⊆ (∅ +o 𝐸))
19	11, 14, 18	rspcedvd 3614	. . . . 5 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∃𝑥 ∈ 𝐸 𝐸 ⊆ (𝑥 +o 𝐸))
20		ssiun 5048	. . . . 5 ⊢ (∃𝑥 ∈ 𝐸 𝐸 ⊆ (𝑥 +o 𝐸) → 𝐸 ⊆ ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸))
21	19, 20	syl 17	. . . 4 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 ⊆ ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸))
22	1	eleq2d 2819	. . . . . . . 8 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (𝑥 ∈ 𝐸 ↔ 𝑥 ∈ (ω ↑o 𝐷)))
23	22	biimpa 477	. . . . . . 7 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → 𝑥 ∈ (ω ↑o 𝐷))
24	6	adantr 481	. . . . . . 7 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → 𝐸 ∈ On)
25	1	adantr 481	. . . . . . . 8 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → 𝐸 = (ω ↑o 𝐷))
26		ssid 4003	. . . . . . . 8 ⊢ 𝐸 ⊆ 𝐸
27	25, 26	eqsstrrdi 4036	. . . . . . 7 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → (ω ↑o 𝐷) ⊆ 𝐸)
28		oaabs2 8644	. . . . . . 7 ⊢ (((𝑥 ∈ (ω ↑o 𝐷) ∧ 𝐸 ∈ On) ∧ (ω ↑o 𝐷) ⊆ 𝐸) → (𝑥 +o 𝐸) = 𝐸)
29	23, 24, 27, 28	syl21anc 836	. . . . . 6 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → (𝑥 +o 𝐸) = 𝐸)
30	29, 26	eqsstrdi 4035	. . . . 5 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → (𝑥 +o 𝐸) ⊆ 𝐸)
31	30	iunssd 5052	. . . 4 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸) ⊆ 𝐸)
32	21, 31	eqssd 3998	. . 3 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 = ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸))
33	6, 6, 32	3jca 1128	. 2 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (𝐸 ∈ On ∧ 𝐸 ∈ On ∧ 𝐸 = ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸)))
34		ofoafg 42089	. 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 593	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))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3070   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  ∪ ciun 4996   × cxp 5673   ↾ cres 5677  Oncon0 6361  ⟶wf 6536  (class class class)co 7405   ∘_f cof 7664  ωcom 7851   +_o coa 8459   ↑_o coe 8461   ↑_m cmap 8816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-omul 8467  df-oexp 8468  df-map 8818

This theorem is referenced by:  ofoafo  42091  ofoacl  42092