Theorem rdgisuc1 6617

Description: One way of describing the value of the recursive definition generator at a successor. There is no condition on the characteristic function 𝐹 other than 𝐹 Fn V. Given that, the resulting expression encompasses both the expected successor term (𝐹‘(rec(𝐹, 𝐴)‘𝐵)) but also terms that correspond to the initial value 𝐴 and to limit ordinals ∪ 𝑥 ∈ 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)).

If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 6618. (Contributed by Jim Kingdon, 9-Jun-2019.)

Hypotheses

Ref	Expression
rdgisuc1.1	⊢ (𝜑 → 𝐹 Fn V)
rdgisuc1.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
rdgisuc1.3	⊢ (𝜑 → 𝐵 ∈ On)

Assertion

Ref	Expression
rdgisuc1	⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rdgisuc1

Step	Hyp	Ref	Expression
1		rdgisuc1.1	. . 3 ⊢ (𝜑 → 𝐹 Fn V)
2		rdgisuc1.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		rdgisuc1.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ On)
4		onsuc 4625	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
5	3, 4	syl 14	. . 3 ⊢ (𝜑 → suc 𝐵 ∈ On)
6		rdgival 6615	. . 3 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ suc 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
7	1, 2, 5, 6	syl3anc 1274	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
8		df-suc 4494	. . . . . . 7 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
9		iuneq1 4006	. . . . . . 7 ⊢ (suc 𝐵 = (𝐵 ∪ {𝐵}) → ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})(𝐹‘(rec(𝐹, 𝐴)‘𝑥)))
10	8, 9	ax-mp 5	. . . . . 6 ⊢ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})(𝐹‘(rec(𝐹, 𝐴)‘𝑥))
11		iunxun 4073	. . . . . 6 ⊢ ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ ∪ 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥)))
12	10, 11	eqtri 2255	. . . . 5 ⊢ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ ∪ 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥)))
13		fveq2 5672	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝐵))
14	13	fveq2d 5676	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
15	14	iunxsng 4069	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
16	15	uneq2d 3375	. . . . 5 ⊢ (𝐵 ∈ On → (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ ∪ 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) = (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
17	12, 16	eqtrid 2279	. . . 4 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
18	17	uneq2d 3375	. . 3 ⊢ (𝐵 ∈ On → (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥))) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
19	3, 18	syl 14	. 2 ⊢ (𝜑 → (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥))) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
20	7, 19	eqtrd 2267	1 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ∪ cun 3211  {csn 3691  ∪ ciun 3993  Oncon0 4486  suc csuc 4488   Fn wfn 5349  ‘cfv 5354  reccrdg 6602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-recs 6538  df-irdg 6603

This theorem is referenced by:  rdgisucinc  6618