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Theorem rdgssun 37344
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 3824 . . . . . . . . . . . 12 𝑥[∅ / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)
2 0ex 5325 . . . . . . . . . . . 12 ∅ ∈ V
3 rzal 4532 . . . . . . . . . . . . 13 (𝑥 = ∅ → ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
4 sbceq1a 3815 . . . . . . . . . . . . 13 (𝑥 = ∅ → (∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ [∅ / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)))
53, 4mpbid 232 . . . . . . . . . . . 12 (𝑥 = ∅ → [∅ / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
61, 2, 5vtoclef 3575 . . . . . . . . . . 11 [∅ / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)
7 vex 3492 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
87elsuc 6465 . . . . . . . . . . . . . . 15 (𝑦 ∈ suc 𝑥 ↔ (𝑦𝑥𝑦 = 𝑥))
9 ssun1 4201 . . . . . . . . . . . . . . . . . . . 20 (rec(𝐹, 𝐴)‘𝑥) ⊆ ((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵)
10 fvex 6933 . . . . . . . . . . . . . . . . . . . . . 22 (rec(𝐹, 𝐴)‘𝑥) ∈ V
11 rdgssun.2 . . . . . . . . . . . . . . . . . . . . . . 23 𝐵 ∈ V
1211csbex 5329 . . . . . . . . . . . . . . . . . . . . . 22 (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵 ∈ V
1310, 12unex 7779 . . . . . . . . . . . . . . . . . . . . 21 ((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵) ∈ V
14 nfcv 2908 . . . . . . . . . . . . . . . . . . . . . 22 𝑤𝐴
15 nfcv 2908 . . . . . . . . . . . . . . . . . . . . . 22 𝑤𝑥
16 rdgssun.1 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝐹 = (𝑤 ∈ V ↦ (𝑤𝐵))
17 nfmpt1 5274 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑤(𝑤 ∈ V ↦ (𝑤𝐵))
1816, 17nfcxfr 2906 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤𝐹
1918, 14nfrdg 8470 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑤rec(𝐹, 𝐴)
2019, 15nffv 6930 . . . . . . . . . . . . . . . . . . . . . . 23 𝑤(rec(𝐹, 𝐴)‘𝑥)
2120nfcsb1 3945 . . . . . . . . . . . . . . . . . . . . . . 23 𝑤(rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵
2220, 21nfun 4193 . . . . . . . . . . . . . . . . . . . . . 22 𝑤((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵)
23 rdgeq1 8467 . . . . . . . . . . . . . . . . . . . . . . 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 3935 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (rec(𝐹, 𝐴)‘𝑥) → 𝐵 = (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵)
2725, 26uneq12d 4192 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = (rec(𝐹, 𝐴)‘𝑥) → (𝑤𝐵) = ((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵))
2814, 15, 22, 24, 27rdgsucmptf 8484 . . . . . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (rec(𝐹, 𝐴)‘suc 𝑥) = ((rec(𝐹, 𝐴)‘𝑥) ∪ (rec(𝐹, 𝐴)‘𝑥) / 𝑤𝐵))
309, 29sseqtrrid 4062 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On → (rec(𝐹, 𝐴)‘𝑥) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥))
31 sstr2 4015 . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ (𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))) → (𝑦𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
35 fveq2 6920 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑥))
3635sseq1d 4040 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑥 → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥) ↔ (rec(𝐹, 𝐴)‘𝑥) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
3730, 36syl5ibrcom 247 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (𝑦 = 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
3837adantr 480 . . . . . . . . . . . . . . . 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 858 . . . . . . . . . . . . . . 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 242 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . 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 3169 . . . . . . . . . . . 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 3805 . . . . . . . . . . . . 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 3492 . . . . . . . . . . . . . . 15 𝑥 ∈ V
4544sucex 7842 . . . . . . . . . . . . . 14 suc 𝑥 ∈ V
46 fveq2 6920 . . . . . . . . . . . . . . . 16 (𝑧 = suc 𝑥 → (rec(𝐹, 𝐴)‘𝑧) = (rec(𝐹, 𝐴)‘suc 𝑥))
4746sseq2d 4041 . . . . . . . . . . . . . . 15 (𝑧 = suc 𝑥 → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧) ↔ (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
4847raleqbi1dv 3346 . . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝑧))
5049sseq2d 4041 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)))
5150raleqbi1dv 3346 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)))
5251cbvabv 2815 . . . . . . . . . . . . . 14 {𝑥 ∣ ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)} = {𝑧 ∣ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)}
5345, 48, 52elab2 3698 . . . . . . . . . . . . 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 275 . . . . . . . . . . . 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, 54imbitrrdi 252 . . . . . . . . . . 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 5070 . . . . . . . . . . . . . . . 16 (𝑦𝑧 → (rec(𝐹, 𝐴)‘𝑦) ⊆ 𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦))
5756adantl 481 . . . . . . . . . . . . . . 15 ((Lim 𝑧𝑦𝑧) → (rec(𝐹, 𝐴)‘𝑦) ⊆ 𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦))
58 vex 3492 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
59 rdglim2a 8489 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, 𝐴)‘𝑧) = 𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦))
6058, 59mpan 689 . . . . . . . . . . . . . . . 16 (Lim 𝑧 → (rec(𝐹, 𝐴)‘𝑧) = 𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦))
6160adantr 480 . . . . . . . . . . . . . . 15 ((Lim 𝑧𝑦𝑧) → (rec(𝐹, 𝐴)‘𝑧) = 𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦))
6257, 61sseqtrrd 4050 . . . . . . . . . . . . . 14 ((Lim 𝑧𝑦𝑧) → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
6362ralrimiva 3152 . . . . . . . . . . . . 13 (Lim 𝑧 → ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
64 df-sbc 3805 . . . . . . . . . . . . . . 15 ([𝑧 / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ 𝑧 ∈ {𝑥 ∣ ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)})
6552eleq2i 2836 . . . . . . . . . . . . . . 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 275 . . . . . . . . . . . . . 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 2721 . . . . . . . . . . . . . 14 (𝑧 ∈ {𝑧 ∣ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)} ↔ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
6866, 67bitri 275 . . . . . . . . . . . . 13 ([𝑧 / 𝑥]𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ ∀𝑦𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
6963, 68sylibr 234 . . . . . . . . . . . 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 7900 . . . . . . . . . 10 (𝑥 ∈ On → ∀𝑦𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
72 rsp 3253 . . . . . . . . . 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 2832 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 ∈ On ↔ 𝑋 ∈ On))
7574adantl 481 . . . . . . . . . 10 ((𝑦 = 𝑌𝑥 = 𝑋) → (𝑥 ∈ On ↔ 𝑋 ∈ On))
76 eleq12 2834 . . . . . . . . . . 11 ((𝑦 = 𝑌𝑥 = 𝑋) → (𝑦𝑥𝑌𝑋))
77 fveq2 6920 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑌))
7877adantr 480 . . . . . . . . . . . 12 ((𝑦 = 𝑌𝑥 = 𝑋) → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑌))
79 fveq2 6920 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝑋))
8079adantl 481 . . . . . . . . . . . 12 ((𝑦 = 𝑌𝑥 = 𝑋) → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝑋))
8178, 80sseq12d 4042 . . . . . . . . . . 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 233 . . . . . . . 8 ((𝑦 = 𝑌𝑥 = 𝑋) → (𝑋 ∈ On → (𝑌𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))))
8584ex 412 . . . . . . 7 (𝑦 = 𝑌 → (𝑥 = 𝑋 → (𝑋 ∈ On → (𝑌𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
8685vtocleg 3565 . . . . . 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 3565 . . . 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 406 1 ((𝑋 ∈ On ∧ 𝑌𝑋) → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 846   = wceq 1537  wcel 2108  {cab 2717  wral 3067  Vcvv 3488  [wsbc 3804  csb 3921  cun 3974  wss 3976  c0 4352   ciun 5015  cmpt 5249  Oncon0 6395  Lim wlim 6396  suc csuc 6397  cfv 6573  reccrdg 8465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466
This theorem is referenced by: (None)
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