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Theorem rdg0 5911
Description: The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
Hypothesis
Ref Expression
rdg.1 A V
Assertion
Ref Expression
rdg0 (rec(𝐹, A)‘∅) = A

Proof of Theorem rdg0
Dummy variables x g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 3874 . . . . 5 V
2 dmeq 4477 . . . . . . . 8 (g = ∅ → dom g = dom ∅)
3 fveq1 5118 . . . . . . . . 9 (g = ∅ → (gx) = (∅‘x))
43fveq2d 5123 . . . . . . . 8 (g = ∅ → (𝐹‘(gx)) = (𝐹‘(∅‘x)))
52, 4iuneq12d 3671 . . . . . . 7 (g = ∅ → x dom g(𝐹‘(gx)) = x dom ∅(𝐹‘(∅‘x)))
65uneq2d 3091 . . . . . 6 (g = ∅ → (A x dom g(𝐹‘(gx))) = (A x dom ∅(𝐹‘(∅‘x))))
7 eqid 2037 . . . . . 6 (g V ↦ (A x dom g(𝐹‘(gx)))) = (g V ↦ (A x dom g(𝐹‘(gx))))
8 rdg.1 . . . . . . 7 A V
9 dm0 4491 . . . . . . . . . 10 dom ∅ = ∅
10 iuneq1 3660 . . . . . . . . . 10 (dom ∅ = ∅ → x dom ∅(𝐹‘(∅‘x)) = x ∅ (𝐹‘(∅‘x)))
119, 10ax-mp 7 . . . . . . . . 9 x dom ∅(𝐹‘(∅‘x)) = x ∅ (𝐹‘(∅‘x))
12 0iun 3704 . . . . . . . . 9 x ∅ (𝐹‘(∅‘x)) = ∅
1311, 12eqtri 2057 . . . . . . . 8 x dom ∅(𝐹‘(∅‘x)) = ∅
1413, 1eqeltri 2107 . . . . . . 7 x dom ∅(𝐹‘(∅‘x)) V
158, 14unex 4141 . . . . . 6 (A x dom ∅(𝐹‘(∅‘x))) V
166, 7, 15fvmpt 5190 . . . . 5 (∅ V → ((g V ↦ (A x dom g(𝐹‘(gx))))‘∅) = (A x dom ∅(𝐹‘(∅‘x))))
171, 16ax-mp 7 . . . 4 ((g V ↦ (A x dom g(𝐹‘(gx))))‘∅) = (A x dom ∅(𝐹‘(∅‘x)))
1817, 15eqeltri 2107 . . 3 ((g V ↦ (A x dom g(𝐹‘(gx))))‘∅) V
19 df-irdg 5894 . . . 4 rec(𝐹, A) = recs((g V ↦ (A x dom g(𝐹‘(gx)))))
2019tfr0 5875 . . 3 (((g V ↦ (A x dom g(𝐹‘(gx))))‘∅) V → (rec(𝐹, A)‘∅) = ((g V ↦ (A x dom g(𝐹‘(gx))))‘∅))
2118, 20ax-mp 7 . 2 (rec(𝐹, A)‘∅) = ((g V ↦ (A x dom g(𝐹‘(gx))))‘∅)
2213uneq2i 3088 . . . 4 (A x dom ∅(𝐹‘(∅‘x))) = (A ∪ ∅)
2317, 22eqtri 2057 . . 3 ((g V ↦ (A x dom g(𝐹‘(gx))))‘∅) = (A ∪ ∅)
24 un0 3245 . . 3 (A ∪ ∅) = A
2523, 24eqtri 2057 . 2 ((g V ↦ (A x dom g(𝐹‘(gx))))‘∅) = A
2621, 25eqtri 2057 1 (rec(𝐹, A)‘∅) = A
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  Vcvv 2551  cun 2909  c0 3218   ciun 3647  cmpt 3808  dom cdm 4287  cfv 4844  reccrdg 5893
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-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219
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-ral 2305  df-rex 2306  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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-id 4020  df-iord 4068  df-on 4070  df-suc 4073  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-res 4299  df-iota 4809  df-fun 4846  df-fn 4847  df-fv 4852  df-recs 5858  df-irdg 5894
This theorem is referenced by:  rdg0g  5912  om0  5970
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