Theorem ofoaf 43344

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 484	. . . 4 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 = (ω ↑o 𝐷))
2		omelon 9599	. . . . 5 ⊢ ω ∈ On
3		simpl 482	. . . . 5 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐷 ∈ On)
4		oecl 8501	. . . . 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 2828	. . 3 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 ∈ On)
7	3, 2	jctil 519	. . . . . . . 8 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (ω ∈ On ∧ 𝐷 ∈ On))
8		peano1 7865	. . . . . . . 8 ⊢ ∅ ∈ ω
9		oen0 8550	. . . . . . . 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 2831	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∅ ∈ 𝐸)
12		oveq1 7394	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 +o 𝐸) = (∅ +o 𝐸))
13	12	sseq2d 3979	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐸 ⊆ (𝑥 +o 𝐸) ↔ 𝐸 ⊆ (∅ +o 𝐸)))
14	13	adantl 481	. . . . . 6 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 = ∅) → (𝐸 ⊆ (𝑥 +o 𝐸) ↔ 𝐸 ⊆ (∅ +o 𝐸)))
15		oa0r 8502	. . . . . . . 8 ⊢ (𝐸 ∈ On → (∅ +o 𝐸) = 𝐸)
16	6, 15	syl 17	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (∅ +o 𝐸) = 𝐸)
17		ssid 3969	. . . . . . 7 ⊢ (∅ +o 𝐸) ⊆ (∅ +o 𝐸)
18	16, 17	eqsstrrdi 3992	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 ⊆ (∅ +o 𝐸))
19	11, 14, 18	rspcedvd 3590	. . . . 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 17	. . . 4 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 ⊆ ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸))
22	1	eleq2d 2814	. . . . . . . 8 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (𝑥 ∈ 𝐸 ↔ 𝑥 ∈ (ω ↑o 𝐷)))
23	22	biimpa 476	. . . . . . 7 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → 𝑥 ∈ (ω ↑o 𝐷))
24	6	adantr 480	. . . . . . 7 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → 𝐸 ∈ On)
25	1	adantr 480	. . . . . . . 8 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → 𝐸 = (ω ↑o 𝐷))
26		ssid 3969	. . . . . . . 8 ⊢ 𝐸 ⊆ 𝐸
27	25, 26	eqsstrrdi 3992	. . . . . . 7 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → (ω ↑o 𝐷) ⊆ 𝐸)
28		oaabs2 8613	. . . . . . 7 ⊢ (((𝑥 ∈ (ω ↑o 𝐷) ∧ 𝐸 ∈ On) ∧ (ω ↑o 𝐷) ⊆ 𝐸) → (𝑥 +o 𝐸) = 𝐸)
29	23, 24, 27, 28	syl21anc 837	. . . . . 6 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → (𝑥 +o 𝐸) = 𝐸)
30	29, 26	eqsstrdi 3991	. . . . 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 3964	. . 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 43343	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  ∪ ciun 4955   × cxp 5636   ↾ cres 5640  Oncon0 6332  ⟶wf 6507  (class class class)co 7387   ∘_f cof 7651  ωcom 7842   +_o coa 8431   ↑_o coe 8433   ↑_m cmap 8799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-omul 8439  df-oexp 8440  df-map 8801

This theorem is referenced by:  ofoafo  43345  ofoacl  43346