Theorem ofoaf 43634

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 9557	. . . . 5 ⊢ ω ∈ On
3		simpl 482	. . . . 5 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐷 ∈ On)
4		oecl 8464	. . . . 5 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ↑o 𝐷) ∈ On)
5	2, 3, 4	sylancr 588	. . . 4 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (ω ↑o 𝐷) ∈ On)
6	1, 5	eqeltrd 2835	. . 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 7831	. . . . . . . 8 ⊢ ∅ ∈ ω
9		oen0 8514	. . . . . . . 8 ⊢ (((ω ∈ On ∧ 𝐷 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐷))
10	7, 8, 9	sylancl 587	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∅ ∈ (ω ↑o 𝐷))
11	10, 1	eleqtrrd 2838	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∅ ∈ 𝐸)
12		oveq1 7365	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 +o 𝐸) = (∅ +o 𝐸))
13	12	sseq2d 3965	. . . . . . 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 8465	. . . . . . . 8 ⊢ (𝐸 ∈ On → (∅ +o 𝐸) = 𝐸)
16	6, 15	syl 17	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (∅ +o 𝐸) = 𝐸)
17		ssid 3955	. . . . . . 7 ⊢ (∅ +o 𝐸) ⊆ (∅ +o 𝐸)
18	16, 17	eqsstrrdi 3978	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 ⊆ (∅ +o 𝐸))
19	11, 14, 18	rspcedvd 3577	. . . . 5 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∃𝑥 ∈ 𝐸 𝐸 ⊆ (𝑥 +o 𝐸))
20		ssiun 5001	. . . . 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 2821	. . . . . . . 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 3955	. . . . . . . 8 ⊢ 𝐸 ⊆ 𝐸
27	25, 26	eqsstrrdi 3978	. . . . . . 7 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → (ω ↑o 𝐷) ⊆ 𝐸)
28		oaabs2 8577	. . . . . . 7 ⊢ (((𝑥 ∈ (ω ↑o 𝐷) ∧ 𝐸 ∈ On) ∧ (ω ↑o 𝐷) ⊆ 𝐸) → (𝑥 +o 𝐸) = 𝐸)
29	23, 24, 27, 28	syl21anc 838	. . . . . 6 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → (𝑥 +o 𝐸) = 𝐸)
30	29, 26	eqsstrdi 3977	. . . . 5 ⊢ (((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) ∧ 𝑥 ∈ 𝐸) → (𝑥 +o 𝐸) ⊆ 𝐸)
31	30	iunssd 5005	. . . 4 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸) ⊆ 𝐸)
32	21, 31	eqssd 3950	. . 3 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → 𝐸 = ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸))
33	6, 6, 32	3jca 1129	. 2 ⊢ ((𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷)) → (𝐸 ∈ On ∧ 𝐸 ∈ On ∧ 𝐸 = ∪ 𝑥 ∈ 𝐸 (𝑥 +o 𝐸)))
34		ofoafg 43633	. 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 594	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 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3059   ∩ cin 3899   ⊆ wss 3900  ∅c0 4284  ∪ ciun 4945   × cxp 5621   ↾ cres 5625  Oncon0 6316  ⟶wf 6487  (class class class)co 7358   ∘_f cof 7620  ωcom 7808   +_o coa 8394   ↑_o coe 8396   ↑_m cmap 8765

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-inf2 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-omul 8402  df-oexp 8403  df-map 8767

This theorem is referenced by:  ofoafo  43635  ofoacl  43636