Theorem ofoaid1 43354

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 766	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → 𝐴 ∈ 𝑉)
2		onss 7764	. . . . . . 7 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
3		sstr 3958	. . . . . . . 8 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ On) → ran 𝐹 ⊆ On)
4	3	expcom 413	. . . . . . 7 ⊢ (𝐵 ⊆ On → (ran 𝐹 ⊆ 𝐵 → ran 𝐹 ⊆ On))
5	2, 4	syl 17	. . . . . 6 ⊢ (𝐵 ∈ On → (ran 𝐹 ⊆ 𝐵 → ran 𝐹 ⊆ On))
6	5	anim2d 612	. . . . 5 ⊢ (𝐵 ∈ On → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ On)))
7		df-f 6518	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
8		df-f 6518	. . . . 5 ⊢ (𝐹:𝐴⟶On ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ On))
9	6, 7, 8	3imtr4g 296	. . . 4 ⊢ (𝐵 ∈ On → (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶On))
10		elmapi 8825	. . . 4 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) → 𝐹:𝐴⟶𝐵)
11	9, 10	impel 505	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → 𝐹:𝐴⟶On)
12	11	adantll 714	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → 𝐹:𝐴⟶On)
13		peano1 7868	. . 3 ⊢ ∅ ∈ ω
14		fnconstg 6751	. . 3 ⊢ (∅ ∈ ω → (𝐴 × {∅}) Fn 𝐴)
15	13, 14	mp1i 13	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → (𝐴 × {∅}) Fn 𝐴)
16		simp2 1137	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐹:𝐴⟶On)
17	16	ffnd 6692	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐹 Fn 𝐴)
18		simp3 1138	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐴 × {∅}) Fn 𝐴)
19		simp1 1136	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐴 ∈ 𝑉)
20		inidm 4193	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
21	17, 18, 19, 19, 20	offn 7669	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐹 ∘f +o (𝐴 × {∅})) Fn 𝐴)
22	17, 18	jca 511	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐹 Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
23	22	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐹 Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
24	19	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝐴 ∈ 𝑉)
25		simpr 484	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
26		fnfvof 7673	. . . . 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 836	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐹 ∘f +o (𝐴 × {∅}))‘𝑎) = ((𝐹‘𝑎) +o ((𝐴 × {∅})‘𝑎)))
28		fvconst2g 7179	. . . . . 6 ⊢ ((∅ ∈ ω ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
29	13, 25, 28	sylancr 587	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
30	29	oveq2d 7406	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) +o ((𝐴 × {∅})‘𝑎)) = ((𝐹‘𝑎) +o ∅))
31	16	ffvelcdmda 7059	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ On)
32		oa0 8483	. . . . 5 ⊢ ((𝐹‘𝑎) ∈ On → ((𝐹‘𝑎) +o ∅) = (𝐹‘𝑎))
33	31, 32	syl 17	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) +o ∅) = (𝐹‘𝑎))
34	27, 30, 33	3eqtrd 2769	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐹 ∘f +o (𝐴 × {∅}))‘𝑎) = (𝐹‘𝑎))
35	21, 17, 34	eqfnfvd 7009	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐹 ∘f +o (𝐴 × {∅})) = 𝐹)
36	1, 12, 15, 35	syl3anc 1373	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → (𝐹 ∘f +o (𝐴 × {∅})) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3917  ∅c0 4299  {csn 4592   × cxp 5639  ran crn 5642  Oncon0 6335   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∘_f cof 7654  ωcom 7845   +_o coa 8434   ↑_m cmap 8802

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-oadd 8441  df-map 8804

This theorem is referenced by: (None)