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Theorem rdgssun 35849
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.)
Hypotheses
Ref Expression
rdgssun.1 𝐹 = (𝑤 ∈ V ↦ (𝑤𝐵))
rdgssun.2 𝐵 ∈ V
Assertion
Ref Expression
rdgssun ((𝑋 ∈ On ∧ 𝑌𝑋) → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))
Distinct variable groups:   𝑤,𝐴   𝑤,𝑌
Allowed substitution hints:   𝐵(𝑤)   𝐹(𝑤)   𝑋(𝑤)

Proof of Theorem rdgssun
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfsbc1v 3759 . . . . . . . . . . . 12 𝑥[∅ / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)
2 0ex 5264 . . . . . . . . . . . 12 ∅ ∈ V
3 rzal 4466 . . . . . . . . . . . . 13 (𝑥 = ∅ → ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
4 sbceq1a 3750 . . . . . . . . . . . . 13 (𝑥 = ∅ → (∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ [∅ / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)))
53, 4mpbid 231 . . . . . . . . . . . 12 (𝑥 = ∅ → [∅ / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
61, 2, 5vtoclef 3515 . . . . . . . . . . 11 [∅ / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)
7 vex 3449 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
87elsuc 6387 . . . . . . . . . . . . . . 15 (𝑦 ∈ suc 𝑥 ↔ (𝑦𝑥𝑦 = 𝑥))
9 ssun1 4132 . . . . . . . . . . . . . . . . . . . 20 (rec(𝐹, 𝐴)‘𝑥) ⊆ ((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵)
10 fvex 6855 . . . . . . . . . . . . . . . . . . . . . 22 (rec(𝐹, 𝐴)‘𝑥) ∈ V
11 rdgssun.2 . . . . . . . . . . . . . . . . . . . . . . 23 𝐵 ∈ V
1211csbex 5268 . . . . . . . . . . . . . . . . . . . . . 22 (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵 ∈ V
1310, 12unex 7680 . . . . . . . . . . . . . . . . . . . . 21 ((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵) ∈ V
14 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . 22 𝑤𝐴
15 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . 22 𝑤𝑥
16 rdgssun.1 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝐹 = (𝑤 ∈ V ↦ (𝑤𝐵))
17 nfmpt1 5213 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑤(𝑤 ∈ V ↦ (𝑤𝐵))
1816, 17nfcxfr 2905 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤𝐹
1918, 14nfrdg 8360 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑤rec(𝐹, 𝐴)
2019, 15nffv 6852 . . . . . . . . . . . . . . . . . . . . . . 23 𝑤(rec(𝐹, 𝐴)‘𝑥)
2120nfcsb1 3879 . . . . . . . . . . . . . . . . . . . . . . 23 𝑤(rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵
2220, 21nfun 4125 . . . . . . . . . . . . . . . . . . . . . 22 𝑤((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵)
23 rdgeq1 8357 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 = (𝑤 ∈ V ↦ (𝑤𝐵)) → rec(𝐹, 𝐴) = rec((𝑤 ∈ V ↦ (𝑤𝐵)), 𝐴))
2416, 23ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 rec(𝐹, 𝐴) = rec((𝑤 ∈ V ↦ (𝑤𝐵)), 𝐴)
25 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (rec(𝐹, 𝐴)‘𝑥) → 𝑤 = (rec(𝐹, 𝐴)‘𝑥))
26 csbeq1a 3869 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (rec(𝐹, 𝐴)‘𝑥) → 𝐵 = (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵)
2725, 26uneq12d 4124 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = (rec(𝐹, 𝐴)‘𝑥) → (𝑤𝐵) = ((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵))
2814, 15, 22, 24, 27rdgsucmptf 8374 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ ((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵) ∈ V) → (rec(𝐹, 𝐴)‘suc 𝑥) = ((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵))
2913, 28mpan2 689 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (rec(𝐹, 𝐴)‘suc 𝑥) = ((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵))
309, 29sseqtrrid 3997 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On → (rec(𝐹, 𝐴)‘𝑥) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥))
31 sstr2 3951 . . . . . . . . . . . . . . . . . . 19 ((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)))
3230, 31syl5com 31 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
3332imim2d 57 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → ((𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)) → (𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥))))
3433imp 407 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ (𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))) → (𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
35 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑥))
3635sseq1d 3975 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑥 → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥) ↔ (rec(𝐹, 𝐴)‘𝑥) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
3730, 36syl5ibrcom 246 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (𝑦 = 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
3837adantr 481 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ (𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))) → (𝑦 = 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
3934, 38jaod 857 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ (𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))) → ((𝑦𝑥𝑦 = 𝑥) → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
408, 39biimtrid 241 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ (𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))) → (𝑦 ∈ suc 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
4140ex 413 . . . . . . . . . . . . 13 (𝑥 ∈ On → ((𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)) → (𝑦 ∈ suc 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥))))
4241ralimdv2 3160 . . . . . . . . . . . 12 (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 3740 . . . . . . . . . . . . 13 ([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 3449 . . . . . . . . . . . . . . 15 𝑥 ∈ V
4544sucex 7741 . . . . . . . . . . . . . 14 suc 𝑥 ∈ V
46 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑧 = suc 𝑥 → (rec(𝐹, 𝐴)‘𝑧) = (rec(𝐹, 𝐴)‘suc 𝑥))
4746sseq2d 3976 . . . . . . . . . . . . . . 15 (𝑧 = suc 𝑥 → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧) ↔ (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
4847raleqbi1dv 3307 . . . . . . . . . . . . . 14 (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 6842 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝑧))
5049sseq2d 3976 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)))
5150raleqbi1dv 3307 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)))
5251cbvabv 2809 . . . . . . . . . . . . . 14 {𝑥 ∣ ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)} = {𝑧 ∣ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)}
5345, 48, 52elab2 3634 . . . . . . . . . . . . 13 (suc 𝑥 ∈ {𝑥 ∣ ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)} ↔ ∀𝑦 ∈ suc 𝑥(rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥))
5443, 53bitri 274 . . . . . . . . . . . 12 ([suc 𝑥 / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ ∀𝑦 ∈ suc 𝑥(rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥))
5542, 54syl6ibr 251 . . . . . . . . . . 11 (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 5007 . . . . . . . . . . . . . . . 16 (𝑦𝑧 → (rec(𝐹, 𝐴)‘𝑦) ⊆ 𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦))
5756adantl 482 . . . . . . . . . . . . . . 15 ((Lim 𝑧𝑦𝑧) → (rec(𝐹, 𝐴)‘𝑦) ⊆ 𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦))
58 vex 3449 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
59 rdglim2a 8379 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, 𝐴)‘𝑧) = 𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦))
6058, 59mpan 688 . . . . . . . . . . . . . . . 16 (Lim 𝑧 → (rec(𝐹, 𝐴)‘𝑧) = 𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦))
6160adantr 481 . . . . . . . . . . . . . . 15 ((Lim 𝑧𝑦𝑧) → (rec(𝐹, 𝐴)‘𝑧) = 𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦))
6257, 61sseqtrrd 3985 . . . . . . . . . . . . . 14 ((Lim 𝑧𝑦𝑧) → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
6362ralrimiva 3143 . . . . . . . . . . . . 13 (Lim 𝑧 → ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
64 df-sbc 3740 . . . . . . . . . . . . . . 15 ([𝑧 / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ 𝑧 ∈ {𝑥 ∣ ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)})
6552eleq2i 2829 . . . . . . . . . . . . . . 15 (𝑧 ∈ {𝑥 ∣ ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)} ↔ 𝑧 ∈ {𝑧 ∣ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)})
6664, 65bitri 274 . . . . . . . . . . . . . 14 ([𝑧 / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ 𝑧 ∈ {𝑧 ∣ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)})
67 abid 2717 . . . . . . . . . . . . . 14 (𝑧 ∈ {𝑧 ∣ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)} ↔ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
6866, 67bitri 274 . . . . . . . . . . . . 13 ([𝑧 / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
6963, 68sylibr 233 . . . . . . . . . . . 12 (Lim 𝑧[𝑧 / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
7069a1d 25 . . . . . . . . . . 11 (Lim 𝑧 → (∀𝑥𝑧𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) → [𝑧 / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)))
716, 55, 70tfindes 7799 . . . . . . . . . 10 (𝑥 ∈ On → ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
72 rsp 3230 . . . . . . . . . 10 (∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) → (𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)))
7371, 72syl 17 . . . . . . . . 9 (𝑥 ∈ On → (𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)))
74 eleq1 2825 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 ∈ On ↔ 𝑋 ∈ On))
7574adantl 482 . . . . . . . . . 10 ((𝑦 = 𝑌𝑥 = 𝑋) → (𝑥 ∈ On ↔ 𝑋 ∈ On))
76 eleq12 2827 . . . . . . . . . . 11 ((𝑦 = 𝑌𝑥 = 𝑋) → (𝑦𝑥𝑌𝑋))
77 fveq2 6842 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑌))
7877adantr 481 . . . . . . . . . . . 12 ((𝑦 = 𝑌𝑥 = 𝑋) → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑌))
79 fveq2 6842 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝑋))
8079adantl 482 . . . . . . . . . . . 12 ((𝑦 = 𝑌𝑥 = 𝑋) → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝑋))
8178, 80sseq12d 3977 . . . . . . . . . . 11 ((𝑦 = 𝑌𝑥 = 𝑋) → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))
8276, 81imbi12d 344 . . . . . . . . . 10 ((𝑦 = 𝑌𝑥 = 𝑋) → ((𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)) ↔ (𝑌𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))))
8375, 82imbi12d 344 . . . . . . . . 9 ((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)))))
8473, 83mpbii 232 . . . . . . . 8 ((𝑦 = 𝑌𝑥 = 𝑋) → (𝑋 ∈ On → (𝑌𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))))
8584ex 413 . . . . . . 7 (𝑦 = 𝑌 → (𝑥 = 𝑋 → (𝑋 ∈ On → (𝑌𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
8685vtocleg 3514 . . . . . 6 (𝑌𝑋 → (𝑥 = 𝑋 → (𝑋 ∈ On → (𝑌𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
8786com12 32 . . . . 5 (𝑥 = 𝑋 → (𝑌𝑋 → (𝑋 ∈ On → (𝑌𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
8887vtocleg 3514 . . . 4 (𝑋 ∈ On → (𝑌𝑋 → (𝑋 ∈ On → (𝑌𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
8988pm2.43b 55 . . 3 (𝑌𝑋 → (𝑋 ∈ On → (𝑌𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))))
9089pm2.43b 55 . 2 (𝑋 ∈ On → (𝑌𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))
9190imp 407 1 ((𝑋 ∈ On ∧ 𝑌𝑋) → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  {cab 2713  wral 3064  Vcvv 3445  [wsbc 3739  csb 3855  cun 3908  wss 3910  c0 4282   ciun 4954  cmpt 5188  Oncon0 6317  Lim wlim 6318  suc csuc 6319  cfv 6496  reccrdg 8355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356
This theorem is referenced by: (None)
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