Theorem sucinc2 6513

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 4411	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
2		ordsucss 4541	. . . . 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 4451	. . 3 ⊢ 𝐵 ⊆ suc 𝐵
6	4, 5	sstrdi 3196	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → suc 𝐴 ⊆ suc 𝐵)
7		onelon 4420	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
8		elex 2774	. . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
9		sucexg 4535	. . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V)
10		suceq 4438	. . . . 5 ⊢ (𝑧 = 𝐴 → suc 𝑧 = suc 𝐴)
11		sucinc.1	. . . . 5 ⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)
12	10, 11	fvmptg 5640	. . . 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 2774	. . . 4 ⊢ (𝐵 ∈ On → 𝐵 ∈ V)
16		sucexg 4535	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ V)
17		suceq 4438	. . . . 5 ⊢ (𝑧 = 𝐵 → suc 𝑧 = suc 𝐵)
18	17, 11	fvmptg 5640	. . . 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 3229	1 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ⊆ wss 3157   ↦ cmpt 4095  Ord word 4398  Oncon0 4399  suc csuc 4401  ‘cfv 5259

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267

This theorem is referenced by: (None)