Theorem ofoaid2 43787

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 767	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → 𝐴 ∈ 𝑉)
2		onss 7739	. . . . . . 7 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
3		sstr 3930	. . . . . . . 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 613	. . . . 5 ⊢ (𝐵 ∈ On → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ On)))
7		df-f 6502	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
8		df-f 6502	. . . . 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 8796	. . . 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 715	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → 𝐹:𝐴⟶On)
13		peano1 7840	. . 3 ⊢ ∅ ∈ ω
14		fnconstg 6728	. . 3 ⊢ (∅ ∈ ω → (𝐴 × {∅}) Fn 𝐴)
15	13, 14	mp1i 13	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → (𝐴 × {∅}) Fn 𝐴)
16		simp3 1139	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐴 × {∅}) Fn 𝐴)
17		simp2 1138	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐹:𝐴⟶On)
18	17	ffnd 6669	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐹 Fn 𝐴)
19		simp1 1137	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐴 ∈ 𝑉)
20		inidm 4167	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
21	16, 18, 19, 19, 20	offn 7644	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → ((𝐴 × {∅}) ∘f +o 𝐹) Fn 𝐴)
22	16, 18	jca 511	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → ((𝐴 × {∅}) Fn 𝐴 ∧ 𝐹 Fn 𝐴))
23	22	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅}) Fn 𝐴 ∧ 𝐹 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 7648	. . . . 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 837	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (((𝐴 × {∅}) ∘f +o 𝐹)‘𝑎) = (((𝐴 × {∅})‘𝑎) +o (𝐹‘𝑎)))
28		fvconst2g 7157	. . . . . 6 ⊢ ((∅ ∈ ω ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
29	13, 25, 28	sylancr 588	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
30	29	oveq1d 7382	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (((𝐴 × {∅})‘𝑎) +o (𝐹‘𝑎)) = (∅ +o (𝐹‘𝑎)))
31	17	ffvelcdmda 7036	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ On)
32		oa0r 8473	. . . . 5 ⊢ ((𝐹‘𝑎) ∈ On → (∅ +o (𝐹‘𝑎)) = (𝐹‘𝑎))
33	31, 32	syl 17	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (∅ +o (𝐹‘𝑎)) = (𝐹‘𝑎))
34	27, 30, 33	3eqtrd 2775	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (((𝐴 × {∅}) ∘f +o 𝐹)‘𝑎) = (𝐹‘𝑎))
35	21, 18, 34	eqfnfvd 6986	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → ((𝐴 × {∅}) ∘f +o 𝐹) = 𝐹)
36	1, 12, 15, 35	syl3anc 1374	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → ((𝐴 × {∅}) ∘f +o 𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889  ∅c0 4273  {csn 4567   × cxp 5629  ran crn 5632  Oncon0 6323   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∘_f cof 7629  ωcom 7817   +_o coa 8402   ↑_m cmap 8773

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 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-oadd 8409  df-map 8775

This theorem is referenced by: (None)