Theorem ofoaid2 44163

Description: Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)

Assertion

Ref	Expression
ofoaid2	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → ((𝐴 × {∅}) ∘f +o 𝐹) = 𝐹)

Proof of Theorem ofoaid2

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 779	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → 𝐴 ∈ 𝑉)
2		onss 7790	. . . . . . 7 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
3		sstr 3946	. . . . . . . 8 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ On) → ran 𝐹 ⊆ On)
4	3	expcom 419	. . . . . . 7 ⊢ (𝐵 ⊆ On → (ran 𝐹 ⊆ 𝐵 → ran 𝐹 ⊆ On))
5	2, 4	syl 18	. . . . . 6 ⊢ (𝐵 ∈ On → (ran 𝐹 ⊆ 𝐵 → ran 𝐹 ⊆ On))
6	5	anim2d 624	. . . . 5 ⊢ (𝐵 ∈ On → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ On)))
7		df-f 6544	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
8		df-f 6544	. . . . 5 ⊢ (𝐹:𝐴⟶On ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ On))
9	6, 7, 8	3imtr4g 299	. . . 4 ⊢ (𝐵 ∈ On → (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶On))
10		elmapi 8852	. . . 4 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) → 𝐹:𝐴⟶𝐵)
11	9, 10	impel 515	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → 𝐹:𝐴⟶On)
12	11	adantll 727	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → 𝐹:𝐴⟶On)
13		peano1 7891	. . 3 ⊢ ∅ ∈ ω
14		fnconstg 6770	. . 3 ⊢ (∅ ∈ ω → (𝐴 × {∅}) Fn 𝐴)
15	13, 14	mp1i 14	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → (𝐴 × {∅}) Fn 𝐴)
16		simp3 1156	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐴 × {∅}) Fn 𝐴)
17		simp2 1155	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐹:𝐴⟶On)
18	17	ffnd 6710	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐹 Fn 𝐴)
19		simp1 1154	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐴 ∈ 𝑉)
20		inidm 4179	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
21	16, 18, 19, 19, 20	offn 7697	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → ((𝐴 × {∅}) ∘f +o 𝐹) Fn 𝐴)
22	16, 18	jca 521	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → ((𝐴 × {∅}) Fn 𝐴 ∧ 𝐹 Fn 𝐴))
23	22	adantr 486	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅}) Fn 𝐴 ∧ 𝐹 Fn 𝐴))
24	19	adantr 486	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝐴 ∈ 𝑉)
25		simpr 490	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
26		fnfvof 7701	. . . . 5 ⊢ ((((𝐴 × {∅}) Fn 𝐴 ∧ 𝐹 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴)) → (((𝐴 × {∅}) ∘f +o 𝐹)‘𝑎) = (((𝐴 × {∅})‘𝑎) +o (𝐹‘𝑎)))
27	23, 24, 25, 26	syl12anc 850	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (((𝐴 × {∅}) ∘f +o 𝐹)‘𝑎) = (((𝐴 × {∅})‘𝑎) +o (𝐹‘𝑎)))
28		fvconst2g 7207	. . . . . 6 ⊢ ((∅ ∈ ω ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
29	13, 25, 28	sylancr 599	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
30	29	oveq1d 7434	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (((𝐴 × {∅})‘𝑎) +o (𝐹‘𝑎)) = (∅ +o (𝐹‘𝑎)))
31	17	ffvelcdmda 7083	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ On)
32		oa0r 8529	. . . . 5 ⊢ ((𝐹‘𝑎) ∈ On → (∅ +o (𝐹‘𝑎)) = (𝐹‘𝑎))
33	31, 32	syl 18	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (∅ +o (𝐹‘𝑎)) = (𝐹‘𝑎))
34	27, 30, 33	3eqtrd 2804	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (((𝐴 × {∅}) ∘f +o 𝐹)‘𝑎) = (𝐹‘𝑎))
35	21, 18, 34	eqfnfvd 7032	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → ((𝐴 × {∅}) ∘f +o 𝐹) = 𝐹)
36	1, 12, 15, 35	syl3anc 1398	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → ((𝐴 × {∅}) ∘f +o 𝐹) = 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906  ∅c0 4286  {csn 4591   × cxp 5661  ran crn 5664  Oncon0 6364   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ∘_f cof 7682  ωcom 7868   +_o coa 8456   ↑_m cmap 8830

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-oadd 8463  df-map 8832

This theorem is used by: (None)