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Theorem rdgssun 35549
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 3736 . . . . . . . . . . . 12 𝑥[∅ / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)
2 0ex 5231 . . . . . . . . . . . 12 ∅ ∈ V
3 rzal 4439 . . . . . . . . . . . . 13 (𝑥 = ∅ → ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
4 sbceq1a 3727 . . . . . . . . . . . . 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 3523 . . . . . . . . . . 11 [∅ / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)
7 vex 3436 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
87elsuc 6335 . . . . . . . . . . . . . . 15 (𝑦 ∈ suc 𝑥 ↔ (𝑦𝑥𝑦 = 𝑥))
9 ssun1 4106 . . . . . . . . . . . . . . . . . . . 20 (rec(𝐹, 𝐴)‘𝑥) ⊆ ((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵)
10 fvex 6787 . . . . . . . . . . . . . . . . . . . . . 22 (rec(𝐹, 𝐴)‘𝑥) ∈ V
11 rdgssun.2 . . . . . . . . . . . . . . . . . . . . . . 23 𝐵 ∈ V
1211csbex 5235 . . . . . . . . . . . . . . . . . . . . . 22 (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵 ∈ V
1310, 12unex 7596 . . . . . . . . . . . . . . . . . . . . 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 5182 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑤(𝑤 ∈ V ↦ (𝑤𝐵))
1816, 17nfcxfr 2905 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤𝐹
1918, 14nfrdg 8245 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑤rec(𝐹, 𝐴)
2019, 15nffv 6784 . . . . . . . . . . . . . . . . . . . . . . 23 𝑤(rec(𝐹, 𝐴)‘𝑥)
2120nfcsb1 3856 . . . . . . . . . . . . . . . . . . . . . . 23 𝑤(rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵
2220, 21nfun 4099 . . . . . . . . . . . . . . . . . . . . . 22 𝑤((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵)
23 rdgeq1 8242 . . . . . . . . . . . . . . . . . . . . . . 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 3846 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (rec(𝐹, 𝐴)‘𝑥) → 𝐵 = (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵)
2725, 26uneq12d 4098 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = (rec(𝐹, 𝐴)‘𝑥) → (𝑤𝐵) = ((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵))
2814, 15, 22, 24, 27rdgsucmptf 8259 . . . . . . . . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (rec(𝐹, 𝐴)‘suc 𝑥) = ((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵))
309, 29sseqtrrid 3974 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On → (rec(𝐹, 𝐴)‘𝑥) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥))
31 sstr2 3928 . . . . . . . . . . . . . . . . . . 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 6774 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑥))
3635sseq1d 3952 . . . . . . . . . . . . . . . . . 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 856 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ (𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))) → ((𝑦𝑥𝑦 = 𝑥) → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
408, 39syl5bi 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 3107 . . . . . . . . . . . 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 3717 . . . . . . . . . . . . 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 3436 . . . . . . . . . . . . . . 15 𝑥 ∈ V
4544sucex 7656 . . . . . . . . . . . . . 14 suc 𝑥 ∈ V
46 fveq2 6774 . . . . . . . . . . . . . . . 16 (𝑧 = suc 𝑥 → (rec(𝐹, 𝐴)‘𝑧) = (rec(𝐹, 𝐴)‘suc 𝑥))
4746sseq2d 3953 . . . . . . . . . . . . . . 15 (𝑧 = suc 𝑥 → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧) ↔ (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
4847raleqbi1dv 3340 . . . . . . . . . . . . . 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 6774 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝑧))
5049sseq2d 3953 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)))
5150raleqbi1dv 3340 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)))
5251cbvabv 2811 . . . . . . . . . . . . . 14 {𝑥 ∣ ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)} = {𝑧 ∣ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)}
5345, 48, 52elab2 3613 . . . . . . . . . . . . 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 4977 . . . . . . . . . . . . . . . 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 3436 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
59 rdglim2a 8264 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, 𝐴)‘𝑧) = 𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦))
6058, 59mpan 687 . . . . . . . . . . . . . . . 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 3962 . . . . . . . . . . . . . 14 ((Lim 𝑧𝑦𝑧) → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
6362ralrimiva 3103 . . . . . . . . . . . . 13 (Lim 𝑧 → ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
64 df-sbc 3717 . . . . . . . . . . . . . . 15 ([𝑧 / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ 𝑧 ∈ {𝑥 ∣ ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)})
6552eleq2i 2830 . . . . . . . . . . . . . . 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 2719 . . . . . . . . . . . . . 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 7709 . . . . . . . . . 10 (𝑥 ∈ On → ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
72 rsp 3131 . . . . . . . . . 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 2826 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 ∈ On ↔ 𝑋 ∈ On))
7574adantl 482 . . . . . . . . . 10 ((𝑦 = 𝑌𝑥 = 𝑋) → (𝑥 ∈ On ↔ 𝑋 ∈ On))
76 eleq12 2828 . . . . . . . . . . 11 ((𝑦 = 𝑌𝑥 = 𝑋) → (𝑦𝑥𝑌𝑋))
77 fveq2 6774 . . . . . . . . . . . . 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 6774 . . . . . . . . . . . . 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 3954 . . . . . . . . . . 11 ((𝑦 = 𝑌𝑥 = 𝑋) → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))
8276, 81imbi12d 345 . . . . . . . . . 10 ((𝑦 = 𝑌𝑥 = 𝑋) → ((𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)) ↔ (𝑌𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))))
8375, 82imbi12d 345 . . . . . . . . 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 3521 . . . . . 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 3521 . . . 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 844   = wceq 1539  wcel 2106  {cab 2715  wral 3064  Vcvv 3432  [wsbc 3716  csb 3832  cun 3885  wss 3887  c0 4256   ciun 4924  cmpt 5157  Oncon0 6266  Lim wlim 6267  suc csuc 6268  cfv 6433  reccrdg 8240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241
This theorem is referenced by: (None)
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