Theorem sucinc2 6679

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 4496	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
2		ordsucss 4626	. . . . 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 4536	. . 3 ⊢ 𝐵 ⊆ suc 𝐵
6	4, 5	sstrdi 3250	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ suc 𝐵)
7		onelon 4505	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
8		elex 2825	. . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
9		sucexg 4620	. . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V)
10		suceq 4523	. . . . 5 ⊢ (𝑧 = 𝐴 → suc 𝑧 = suc 𝐴)
11		sucinc.1	. . . . 5 ⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)
12	10, 11	fvmptg 5753	. . . 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 2825	. . . 4 ⊢ (𝐵 ∈ On → 𝐵 ∈ V)
16		sucexg 4620	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ V)
17		suceq 4523	. . . . 5 ⊢ (𝑧 = 𝐵 → suc 𝑧 = suc 𝐵)
18	17, 11	fvmptg 5753	. . . 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 3283	1 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  Vcvv 2813   ⊆ wss 3211   ↦ cmpt 4171  Ord word 4483  Oncon0 4484  suc csuc 4486  ‘cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360

This theorem is referenced by: (None)