ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rdgisucinc Structured version   GIF version

Theorem rdgisucinc 5912
Description: Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 5983 and omsuc 5990. (Contributed by Jim Kingdon, 29-Aug-2019.)

Hypotheses
Ref Expression
rdgisuc1.1 (φ𝐹 Fn V)
rdgisuc1.2 (φA 𝑉)
rdgisuc1.3 (φB On)
rdgisucinc.inc (φx x ⊆ (𝐹x))
Assertion
Ref Expression
rdgisucinc (φ → (rec(𝐹, A)‘suc B) = (𝐹‘(rec(𝐹, A)‘B)))
Distinct variable groups:   x,𝐹   x,A   x,B   x,𝑉
Allowed substitution hint:   φ(x)

Proof of Theorem rdgisucinc
StepHypRef Expression
1 rdgisuc1.1 . . . 4 (φ𝐹 Fn V)
2 rdgisuc1.2 . . . 4 (φA 𝑉)
3 rdgisuc1.3 . . . 4 (φB On)
41, 2, 3rdgisuc1 5911 . . 3 (φ → (rec(𝐹, A)‘suc B) = (A ∪ ( x B (𝐹‘(rec(𝐹, A)‘x)) ∪ (𝐹‘(rec(𝐹, A)‘B)))))
5 unass 3094 . . 3 ((A x B (𝐹‘(rec(𝐹, A)‘x))) ∪ (𝐹‘(rec(𝐹, A)‘B))) = (A ∪ ( x B (𝐹‘(rec(𝐹, A)‘x)) ∪ (𝐹‘(rec(𝐹, A)‘B))))
64, 5syl6eqr 2087 . 2 (φ → (rec(𝐹, A)‘suc B) = ((A x B (𝐹‘(rec(𝐹, A)‘x))) ∪ (𝐹‘(rec(𝐹, A)‘B))))
7 rdgival 5909 . . . 4 ((𝐹 Fn V A 𝑉 B On) → (rec(𝐹, A)‘B) = (A x B (𝐹‘(rec(𝐹, A)‘x))))
81, 2, 3, 7syl3anc 1134 . . 3 (φ → (rec(𝐹, A)‘B) = (A x B (𝐹‘(rec(𝐹, A)‘x))))
98uneq1d 3090 . 2 (φ → ((rec(𝐹, A)‘B) ∪ (𝐹‘(rec(𝐹, A)‘B))) = ((A x B (𝐹‘(rec(𝐹, A)‘x))) ∪ (𝐹‘(rec(𝐹, A)‘B))))
10 rdgexggg 5904 . . . . 5 ((𝐹 Fn V A 𝑉 B On) → (rec(𝐹, A)‘B) V)
111, 2, 3, 10syl3anc 1134 . . . 4 (φ → (rec(𝐹, A)‘B) V)
12 rdgisucinc.inc . . . 4 (φx x ⊆ (𝐹x))
13 id 19 . . . . . 6 (x = (rec(𝐹, A)‘B) → x = (rec(𝐹, A)‘B))
14 fveq2 5121 . . . . . 6 (x = (rec(𝐹, A)‘B) → (𝐹x) = (𝐹‘(rec(𝐹, A)‘B)))
1513, 14sseq12d 2968 . . . . 5 (x = (rec(𝐹, A)‘B) → (x ⊆ (𝐹x) ↔ (rec(𝐹, A)‘B) ⊆ (𝐹‘(rec(𝐹, A)‘B))))
1615spcgv 2634 . . . 4 ((rec(𝐹, A)‘B) V → (x x ⊆ (𝐹x) → (rec(𝐹, A)‘B) ⊆ (𝐹‘(rec(𝐹, A)‘B))))
1711, 12, 16sylc 56 . . 3 (φ → (rec(𝐹, A)‘B) ⊆ (𝐹‘(rec(𝐹, A)‘B)))
18 ssequn1 3107 . . 3 ((rec(𝐹, A)‘B) ⊆ (𝐹‘(rec(𝐹, A)‘B)) ↔ ((rec(𝐹, A)‘B) ∪ (𝐹‘(rec(𝐹, A)‘B))) = (𝐹‘(rec(𝐹, A)‘B)))
1917, 18sylib 127 . 2 (φ → ((rec(𝐹, A)‘B) ∪ (𝐹‘(rec(𝐹, A)‘B))) = (𝐹‘(rec(𝐹, A)‘B)))
206, 9, 193eqtr2d 2075 1 (φ → (rec(𝐹, A)‘suc B) = (𝐹‘(rec(𝐹, A)‘B)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240   = wceq 1242   wcel 1390  Vcvv 2551  cun 2909  wss 2911   ciun 3648  Oncon0 4066  suc csuc 4068   Fn wfn 4840  cfv 4845  reccrdg 5896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-recs 5861  df-irdg 5897
This theorem is referenced by:  frecrdg  5931
  Copyright terms: Public domain W3C validator