Theorem rdgellim 37870

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 6867	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝐶))
2	1	eleq2d 2848	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦) ↔ 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶)))
3	2	rspcev 3581	. . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶)) → ∃𝑦 ∈ 𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦))
4	3	ex 416	. . . 4 ⊢ (𝐶 ∈ 𝐵 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → ∃𝑦 ∈ 𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦)))
5		eliun 4953	. . . 4 ⊢ (𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦) ↔ ∃𝑦 ∈ 𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦))
6	4, 5	imbitrrdi 254	. . 3 ⊢ (𝐶 ∈ 𝐵 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦)))
7	6	adantl 485	. 2 ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦)))
8		rdglim2a 8404	. . . 4 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦))
9	8	eleq2d 2848	. . 3 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵) ↔ 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦)))
10	9	adantr 484	. 2 ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵) ↔ 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦)))
11	7, 10	sylibrd 261	1 ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∃wrex 3086  ∪ ciun 4949  Oncon0 6346  Lim wlim 6347  ‘cfv 6521  reccrdg 8380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381

This theorem is referenced by:  rdglimss  37871  exrecfnlem  37873