Theorem sucinc2 6425

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 4360	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
2		ordsucss 4488	. . . . 5 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
3	1, 2	syl 14	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
4	3	imp 123	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ 𝐵)
5		sssucid 4400	. . 3 ⊢ 𝐵 ⊆ suc 𝐵
6	4, 5	sstrdi 3159	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ suc 𝐵)
7		onelon 4369	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
8		elex 2741	. . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
9		sucexg 4482	. . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V)
10		suceq 4387	. . . . 5 ⊢ (𝑧 = 𝐴 → suc 𝑧 = suc 𝐴)
11		sucinc.1	. . . . 5 ⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)
12	10, 11	fvmptg 5572	. . . 4 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 ∈ V) → (𝐹‘𝐴) = suc 𝐴)
13	8, 9, 12	syl2anc 409	. . 3 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = suc 𝐴)
14	7, 13	syl 14	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = suc 𝐴)
15		elex 2741	. . . 4 ⊢ (𝐵 ∈ On → 𝐵 ∈ V)
16		sucexg 4482	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ V)
17		suceq 4387	. . . . 5 ⊢ (𝑧 = 𝐵 → suc 𝑧 = suc 𝐵)
18	17, 11	fvmptg 5572	. . . 4 ⊢ ((𝐵 ∈ V ∧ suc 𝐵 ∈ V) → (𝐹‘𝐵) = suc 𝐵)
19	15, 16, 18	syl2anc 409	. . 3 ⊢ (𝐵 ∈ On → (𝐹‘𝐵) = suc 𝐵)
20	19	adantr 274	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐵) = suc 𝐵)
21	6, 14, 20	3sstr4d 3192	1 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  Vcvv 2730   ⊆ wss 3121   ↦ cmpt 4050  Ord word 4347  Oncon0 4348  suc csuc 4350  ‘cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206

This theorem is referenced by: (None)