Theorem sucinc2 6501

Description: Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)

Hypothesis

Ref	Expression
sucinc.1	⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)

Assertion

Ref	Expression
sucinc2	⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))

Distinct variable groups:   𝑧,𝐴   𝑧,𝐵

Allowed substitution hint:   𝐹(𝑧)

Proof of Theorem sucinc2

Step	Hyp	Ref	Expression
1		eloni 4407	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
2		ordsucss 4537	. . . . 5 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
3	1, 2	syl 14	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
4	3	imp 124	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ 𝐵)
5		sssucid 4447	. . 3 ⊢ 𝐵 ⊆ suc 𝐵
6	4, 5	sstrdi 3192	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ suc 𝐵)
7		onelon 4416	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
8		elex 2771	. . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
9		sucexg 4531	. . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V)
10		suceq 4434	. . . . 5 ⊢ (𝑧 = 𝐴 → suc 𝑧 = suc 𝐴)
11		sucinc.1	. . . . 5 ⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)
12	10, 11	fvmptg 5634	. . . 4 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 ∈ V) → (𝐹‘𝐴) = suc 𝐴)
13	8, 9, 12	syl2anc 411	. . 3 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = suc 𝐴)
14	7, 13	syl 14	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = suc 𝐴)
15		elex 2771	. . . 4 ⊢ (𝐵 ∈ On → 𝐵 ∈ V)
16		sucexg 4531	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ V)
17		suceq 4434	. . . . 5 ⊢ (𝑧 = 𝐵 → suc 𝑧 = suc 𝐵)
18	17, 11	fvmptg 5634	. . . 4 ⊢ ((𝐵 ∈ V ∧ suc 𝐵 ∈ V) → (𝐹‘𝐵) = suc 𝐵)
19	15, 16, 18	syl2anc 411	. . 3 ⊢ (𝐵 ∈ On → (𝐹‘𝐵) = suc 𝐵)
20	19	adantr 276	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐵) = suc 𝐵)
21	6, 14, 20	3sstr4d 3225	1 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3154   ↦ cmpt 4091  Ord word 4394  Oncon0 4395  suc csuc 4397  ‘cfv 5255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263

This theorem is referenced by: (None)