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Theorem rdgisucinc 5892
Description: Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 5959 and omsuc 5966. (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 5891 . . 3 (φ → (rec(𝐹, A)‘suc B) = (A ∪ ( x B (𝐹‘(rec(𝐹, A)‘x)) ∪ (𝐹‘(rec(𝐹, A)‘B)))))
5 unass 3077 . . 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 2072 . 2 (φ → (rec(𝐹, A)‘suc B) = ((A x B (𝐹‘(rec(𝐹, A)‘x))) ∪ (𝐹‘(rec(𝐹, A)‘B))))
7 rdgival 5889 . . . 4 ((𝐹 Fn V A 𝑉 B On) → (rec(𝐹, A)‘B) = (A x B (𝐹‘(rec(𝐹, A)‘x))))
81, 2, 3, 7syl3anc 1121 . . 3 (φ → (rec(𝐹, A)‘B) = (A x B (𝐹‘(rec(𝐹, A)‘x))))
98uneq1d 3073 . 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 5884 . . . . 5 ((𝐹 Fn V A 𝑉 B On) → (rec(𝐹, A)‘B) V)
111, 2, 3, 10syl3anc 1121 . . . 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 5103 . . . . . 6 (x = (rec(𝐹, A)‘B) → (𝐹x) = (𝐹‘(rec(𝐹, A)‘B)))
1513, 14sseq12d 2951 . . . . 5 (x = (rec(𝐹, A)‘B) → (x ⊆ (𝐹x) ↔ (rec(𝐹, A)‘B) ⊆ (𝐹‘(rec(𝐹, A)‘B))))
1615spcgv 2617 . . . 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 3090 . . 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 2060 1 (φ → (rec(𝐹, A)‘suc B) = (𝐹‘(rec(𝐹, A)‘B)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1226   = wceq 1228   wcel 1374  Vcvv 2535  cun 2892  wss 2894   ciun 3631  Oncon0 4049  suc csuc 4051   Fn wfn 4824  cfv 4829  reccrdg 5877
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-recs 5842  df-irdg 5878
This theorem is referenced by:  frecrdg  5908
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