Theorem rdgellim 37581

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 6834	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝐶))
2	1	eleq2d 2822	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦) ↔ 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶)))
3	2	rspcev 3576	. . . . 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 4950	. . . 4 ⊢ (𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦) ↔ ∃𝑦 ∈ 𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦))
6	4, 5	imbitrrdi 252	. . 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 8364	. . . 4 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦))
9	8	eleq2d 2822	. . 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  ∪ ciun 4946  Oncon0 6317  Lim wlim 6318  ‘cfv 6492  reccrdg 8340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341

This theorem is referenced by:  rdglimss  37582  exrecfnlem  37584