Metamath Proof Explorer


Theorem rdgssun

Description: In a recursive definition where each step expands on the previous one using a union, every previous step is a subset of every later step. (Contributed by ML, 1-Apr-2022)

Ref Expression
Hypotheses rdgssun.1 𝐹 = ( 𝑤 ∈ V ↦ ( 𝑤𝐵 ) )
rdgssun.2 𝐵 ∈ V
Assertion rdgssun ( ( 𝑋 ∈ On ∧ 𝑌𝑋 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 rdgssun.1 𝐹 = ( 𝑤 ∈ V ↦ ( 𝑤𝐵 ) )
2 rdgssun.2 𝐵 ∈ V
3 nfsbc1v 𝑥 [ ∅ / 𝑥 ]𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 )
4 0ex ∅ ∈ V
5 rzal ( 𝑥 = ∅ → ∀ 𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) )
6 sbceq1a ( 𝑥 = ∅ → ( ∀ 𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ [ ∅ / 𝑥 ]𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) )
7 5 6 mpbid ( 𝑥 = ∅ → [ ∅ / 𝑥 ]𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) )
8 3 4 7 vtoclef [ ∅ / 𝑥 ]𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 )
9 vex 𝑦 ∈ V
10 9 elsuc ( 𝑦 ∈ suc 𝑥 ↔ ( 𝑦𝑥𝑦 = 𝑥 ) )
11 ssun1 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ⊆ ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 𝐵 )
12 fvex ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∈ V
13 2 csbex ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 𝐵 ∈ V
14 12 13 unex ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 𝐵 ) ∈ V
15 nfcv 𝑤 𝐴
16 nfcv 𝑤 𝑥
17 nfmpt1 𝑤 ( 𝑤 ∈ V ↦ ( 𝑤𝐵 ) )
18 1 17 nfcxfr 𝑤 𝐹
19 18 15 nfrdg 𝑤 rec ( 𝐹 , 𝐴 )
20 19 16 nffv 𝑤 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 )
21 20 nfcsb1 𝑤 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 𝐵
22 20 21 nfun 𝑤 ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 𝐵 )
23 rdgeq1 ( 𝐹 = ( 𝑤 ∈ V ↦ ( 𝑤𝐵 ) ) → rec ( 𝐹 , 𝐴 ) = rec ( ( 𝑤 ∈ V ↦ ( 𝑤𝐵 ) ) , 𝐴 ) )
24 1 23 ax-mp rec ( 𝐹 , 𝐴 ) = rec ( ( 𝑤 ∈ V ↦ ( 𝑤𝐵 ) ) , 𝐴 )
25 id ( 𝑤 = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → 𝑤 = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) )
26 csbeq1a ( 𝑤 = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → 𝐵 = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 𝐵 )
27 25 26 uneq12d ( 𝑤 = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → ( 𝑤𝐵 ) = ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 𝐵 ) )
28 15 16 22 24 27 rdgsucmptf ( ( 𝑥 ∈ On ∧ ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 𝐵 ) ∈ V ) → ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) = ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 𝐵 ) )
29 14 28 mpan2 ( 𝑥 ∈ On → ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) = ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 𝐵 ) )
30 11 29 sseqtrrid ( 𝑥 ∈ On → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) )
31 sstr2 ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) )
32 30 31 syl5com ( 𝑥 ∈ On → ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) )
33 32 imim2d ( 𝑥 ∈ On → ( ( 𝑦𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) → ( 𝑦𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) )
34 33 imp ( ( 𝑥 ∈ On ∧ ( 𝑦𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) → ( 𝑦𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) )
35 fveq2 ( 𝑦 = 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) )
36 35 sseq1d ( 𝑦 = 𝑥 → ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ↔ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) )
37 30 36 syl5ibrcom ( 𝑥 ∈ On → ( 𝑦 = 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) )
38 37 adantr ( ( 𝑥 ∈ On ∧ ( 𝑦𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) → ( 𝑦 = 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) )
39 34 38 jaod ( ( 𝑥 ∈ On ∧ ( 𝑦𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) → ( ( 𝑦𝑥𝑦 = 𝑥 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) )
40 10 39 syl5bi ( ( 𝑥 ∈ On ∧ ( 𝑦𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) → ( 𝑦 ∈ suc 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) )
41 40 ex ( 𝑥 ∈ On → ( ( 𝑦𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) → ( 𝑦 ∈ suc 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) )
42 41 ralimdv2 ( 𝑥 ∈ On → ( ∀ 𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → ∀ 𝑦 ∈ suc 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) )
43 df-sbc ( [ suc 𝑥 / 𝑥 ]𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ suc 𝑥 ∈ { 𝑥 ∣ ∀ 𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) } )
44 vex 𝑥 ∈ V
45 44 sucex suc 𝑥 ∈ V
46 fveq2 ( 𝑧 = suc 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) = ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) )
47 46 sseq2d ( 𝑧 = suc 𝑥 → ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) ↔ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) )
48 47 raleqbi1dv ( 𝑧 = suc 𝑥 → ( ∀ 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) ↔ ∀ 𝑦 ∈ suc 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) )
49 fveq2 ( 𝑥 = 𝑧 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) )
50 49 sseq2d ( 𝑥 = 𝑧 → ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) ) )
51 50 raleqbi1dv ( 𝑥 = 𝑧 → ( ∀ 𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ ∀ 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) ) )
52 51 cbvabv { 𝑥 ∣ ∀ 𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) } = { 𝑧 ∣ ∀ 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) }
53 45 48 52 elab2 ( suc 𝑥 ∈ { 𝑥 ∣ ∀ 𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) } ↔ ∀ 𝑦 ∈ suc 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) )
54 43 53 bitri ( [ suc 𝑥 / 𝑥 ]𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ ∀ 𝑦 ∈ suc 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) )
55 42 54 syl6ibr ( 𝑥 ∈ On → ( ∀ 𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → [ suc 𝑥 / 𝑥 ]𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) )
56 ssiun2 ( 𝑦𝑧 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) )
57 56 adantl ( ( Lim 𝑧𝑦𝑧 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) )
58 vex 𝑧 ∈ V
59 rdglim2a ( ( 𝑧 ∈ V ∧ Lim 𝑧 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) = 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) )
60 58 59 mpan ( Lim 𝑧 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) = 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) )
61 60 adantr ( ( Lim 𝑧𝑦𝑧 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) = 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) )
62 57 61 sseqtrrd ( ( Lim 𝑧𝑦𝑧 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) )
63 62 ralrimiva ( Lim 𝑧 → ∀ 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) )
64 df-sbc ( [ 𝑧 / 𝑥 ]𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ 𝑧 ∈ { 𝑥 ∣ ∀ 𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) } )
65 52 eleq2i ( 𝑧 ∈ { 𝑥 ∣ ∀ 𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) } ↔ 𝑧 ∈ { 𝑧 ∣ ∀ 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) } )
66 64 65 bitri ( [ 𝑧 / 𝑥 ]𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ 𝑧 ∈ { 𝑧 ∣ ∀ 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) } )
67 abid ( 𝑧 ∈ { 𝑧 ∣ ∀ 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) } ↔ ∀ 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) )
68 66 67 bitri ( [ 𝑧 / 𝑥 ]𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ ∀ 𝑦𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) )
69 63 68 sylibr ( Lim 𝑧[ 𝑧 / 𝑥 ]𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) )
70 69 a1d ( Lim 𝑧 → ( ∀ 𝑥𝑧𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → [ 𝑧 / 𝑥 ]𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) )
71 8 55 70 tfindes ( 𝑥 ∈ On → ∀ 𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) )
72 rsp ( ∀ 𝑦𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → ( 𝑦𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) )
73 71 72 syl ( 𝑥 ∈ On → ( 𝑦𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) )
74 eleq1 ( 𝑥 = 𝑋 → ( 𝑥 ∈ On ↔ 𝑋 ∈ On ) )
75 74 adantl ( ( 𝑦 = 𝑌𝑥 = 𝑋 ) → ( 𝑥 ∈ On ↔ 𝑋 ∈ On ) )
76 eleq12 ( ( 𝑦 = 𝑌𝑥 = 𝑋 ) → ( 𝑦𝑥𝑌𝑋 ) )
77 fveq2 ( 𝑦 = 𝑌 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) )
78 77 adantr ( ( 𝑦 = 𝑌𝑥 = 𝑋 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) )
79 fveq2 ( 𝑥 = 𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) )
80 79 adantl ( ( 𝑦 = 𝑌𝑥 = 𝑋 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) )
81 78 80 sseq12d ( ( 𝑦 = 𝑌𝑥 = 𝑋 ) → ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) )
82 76 81 imbi12d ( ( 𝑦 = 𝑌𝑥 = 𝑋 ) → ( ( 𝑦𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ↔ ( 𝑌𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) )
83 75 82 imbi12d ( ( 𝑦 = 𝑌𝑥 = 𝑋 ) → ( ( 𝑥 ∈ On → ( 𝑦𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) ↔ ( 𝑋 ∈ On → ( 𝑌𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) ) )
84 73 83 mpbii ( ( 𝑦 = 𝑌𝑥 = 𝑋 ) → ( 𝑋 ∈ On → ( 𝑌𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) )
85 84 ex ( 𝑦 = 𝑌 → ( 𝑥 = 𝑋 → ( 𝑋 ∈ On → ( 𝑌𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) ) )
86 85 vtocleg ( 𝑌𝑋 → ( 𝑥 = 𝑋 → ( 𝑋 ∈ On → ( 𝑌𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) ) )
87 86 com12 ( 𝑥 = 𝑋 → ( 𝑌𝑋 → ( 𝑋 ∈ On → ( 𝑌𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) ) )
88 87 vtocleg ( 𝑋 ∈ On → ( 𝑌𝑋 → ( 𝑋 ∈ On → ( 𝑌𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) ) )
89 88 pm2.43b ( 𝑌𝑋 → ( 𝑋 ∈ On → ( 𝑌𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) )
90 89 pm2.43b ( 𝑋 ∈ On → ( 𝑌𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) )
91 90 imp ( ( 𝑋 ∈ On ∧ 𝑌𝑋 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) )