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Theorem tfr2a 5874
 Description: A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.)
Hypothesis
Ref Expression
tfr.1 𝐹 = recs(𝐺)
Assertion
Ref Expression
tfr2a (A dom 𝐹 → (𝐹A) = (𝐺‘(𝐹A)))

Proof of Theorem tfr2a
Dummy variables x f y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2037 . . . 4 {fx On (f Fn x y x (fy) = (𝐺‘(fy)))} = {fx On (f Fn x y x (fy) = (𝐺‘(fy)))}
21tfrlem9 5873 . . 3 (A dom recs(𝐺) → (recs(𝐺)‘A) = (𝐺‘(recs(𝐺) ↾ A)))
3 tfr.1 . . . 4 𝐹 = recs(𝐺)
43dmeqi 4478 . . 3 dom 𝐹 = dom recs(𝐺)
52, 4eleq2s 2129 . 2 (A dom 𝐹 → (recs(𝐺)‘A) = (𝐺‘(recs(𝐺) ↾ A)))
63fveq1i 5120 . 2 (𝐹A) = (recs(𝐺)‘A)
73reseq1i 4550 . . 3 (𝐹A) = (recs(𝐺) ↾ A)
87fveq2i 5122 . 2 (𝐺‘(𝐹A)) = (𝐺‘(recs(𝐺) ↾ A))
95, 6, 83eqtr4g 2094 1 (A dom 𝐹 → (𝐹A) = (𝐺‘(𝐹A)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  Oncon0 4065  dom cdm 4287   ↾ cres 4289   Fn wfn 4839  ‘cfv 4844  recscrecs 5857 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-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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934  ax-setind 4219 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  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-un 2916  df-in 2918  df-ss 2925  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-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 This theorem is referenced by:  tfr0  5875  tfri2d  5888  tfri2  5890
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