Theorem rdgellim 35547

Description: Elementhood in a recursive definition at a limit ordinal. (Contributed by ML, 30-Mar-2022.)

Assertion

Ref	Expression
rdgellim	⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵)))

Proof of Theorem rdgellim

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6774	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝐶))
2	1	eleq2d 2824	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦) ↔ 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶)))
3	2	rspcev 3561	. . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶)) → ∃𝑦 ∈ 𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦))
4	3	ex 413	. . . 4 ⊢ (𝐶 ∈ 𝐵 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → ∃𝑦 ∈ 𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦)))
5		eliun 4928	. . . 4 ⊢ (𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦) ↔ ∃𝑦 ∈ 𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦))
6	4, 5	syl6ibr 251	. . 3 ⊢ (𝐶 ∈ 𝐵 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦)))
7	6	adantl 482	. 2 ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦)))
8		rdglim2a 8264	. . . 4 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦))
9	8	eleq2d 2824	. . 3 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵) ↔ 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦)))
10	9	adantr 481	. 2 ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵) ↔ 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦)))
11	7, 10	sylibrd 258	1 ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  ∪ ciun 4924  Oncon0 6266  Lim wlim 6267  ‘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:  rdglimss  35548  exrecfnlem  35550