Theorem rdgellim 36764

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 6885	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝐶))
2	1	eleq2d 2813	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦) ↔ 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶)))
3	2	rspcev 3606	. . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶)) → ∃𝑦 ∈ 𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦))
4	3	ex 412	. . . 4 ⊢ (𝐶 ∈ 𝐵 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → ∃𝑦 ∈ 𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦)))
5		eliun 4994	. . . 4 ⊢ (𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦) ↔ ∃𝑦 ∈ 𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦))
6	4, 5	imbitrrdi 251	. . 3 ⊢ (𝐶 ∈ 𝐵 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦)))
7	6	adantl 481	. 2 ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦)))
8		rdglim2a 8434	. . . 4 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦))
9	8	eleq2d 2813	. . 3 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵) ↔ 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦)))
10	9	adantr 480	. 2 ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵) ↔ 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦)))
11	7, 10	sylibrd 259	1 ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∃wrex 3064  ∪ ciun 4990  Oncon0 6358  Lim wlim 6359  ‘cfv 6537  reccrdg 8410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411

This theorem is referenced by:  rdglimss  36765  exrecfnlem  36767