Theorem ofoaid1 44552

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

Assertion

Ref	Expression
ofoaid1	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → (𝐹 ∘f +o (𝐴 × {∅})) = 𝐹)

Proof of Theorem ofoaid1

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 7823	. . . . . . 7 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
3		sstr 3921	. . . . . . . 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 6562	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
8		df-f 6562	. . . . 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 8897	. . . 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 7924	. . 3 ⊢ ∅ ∈ ω
14		fnconstg 6792	. . 3 ⊢ (∅ ∈ ω → (𝐴 × {∅}) Fn 𝐴)
15	13, 14	mp1i 14	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → (𝐴 × {∅}) Fn 𝐴)
16		simp2 1155	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐹:𝐴⟶On)
17	16	ffnd 6732	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐹 Fn 𝐴)
18		simp3 1156	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐴 × {∅}) Fn 𝐴)
19		simp1 1154	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐴 ∈ 𝑉)
20		inidm 4126	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
21	17, 18, 19, 19, 20	offn 7730	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐹 ∘f +o (𝐴 × {∅})) Fn 𝐴)
22	17, 18	jca 521	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐹 Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
23	22	adantr 486	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐹 Fn 𝐴 ∧ (𝐴 × {∅}) 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 7734	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴)) → ((𝐹 ∘f +o (𝐴 × {∅}))‘𝑎) = ((𝐹‘𝑎) +o ((𝐴 × {∅})‘𝑎)))
27	23, 24, 25, 26	syl12anc 850	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐹 ∘f +o (𝐴 × {∅}))‘𝑎) = ((𝐹‘𝑎) +o ((𝐴 × {∅})‘𝑎)))
28		fvconst2g 7231	. . . . . 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	oveq2d 7459	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) +o ((𝐴 × {∅})‘𝑎)) = ((𝐹‘𝑎) +o ∅))
31	16	ffvelcdmda 7108	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ On)
32		oa0 8552	. . . . 5 ⊢ ((𝐹‘𝑎) ∈ On → ((𝐹‘𝑎) +o ∅) = (𝐹‘𝑎))
33	31, 32	syl 18	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) +o ∅) = (𝐹‘𝑎))
34	27, 30, 33	3eqtrd 2802	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐹 ∘f +o (𝐴 × {∅}))‘𝑎) = (𝐹‘𝑎))
35	21, 17, 34	eqfnfvd 7055	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐹 ∘f +o (𝐴 × {∅})) = 𝐹)
36	1, 12, 15, 35	syl3anc 1399	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → (𝐹 ∘f +o (𝐴 × {∅})) = 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  ∅c0 4281  {csn 4588   × cxp 5653  ran crn 5656  Oncon0 6380   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443   ∘_f cof 7715  ωcom 7901   +_o coa 8501   ↑_m cmap 8875

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-oadd 8508  df-map 8877

This theorem is used by: (None)