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Theorem tfr1a 8390
Description: A weak version of tfr1 8393 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
Hypothesis
Ref Expression
tfr.1 𝐹 = recs(𝐺)
Assertion
Ref Expression
tfr1a (Fun 𝐹 ∧ Lim dom 𝐹)

Proof of Theorem tfr1a
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem7 8379 . . 3 Fun recs(𝐺)
3 tfr.1 . . . 4 𝐹 = recs(𝐺)
43funeqi 6566 . . 3 (Fun 𝐹 ↔ Fun recs(𝐺))
52, 4mpbir 230 . 2 Fun 𝐹
61tfrlem16 8389 . . 3 Lim dom recs(𝐺)
73dmeqi 5902 . . . 4 dom 𝐹 = dom recs(𝐺)
8 limeq 6373 . . . 4 (dom 𝐹 = dom recs(𝐺) → (Lim dom 𝐹 ↔ Lim dom recs(𝐺)))
97, 8ax-mp 5 . . 3 (Lim dom 𝐹 ↔ Lim dom recs(𝐺))
106, 9mpbir 230 . 2 Lim dom 𝐹
115, 10pm3.2i 471 1 (Fun 𝐹 ∧ Lim dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  {cab 2709  wral 3061  wrex 3070  dom cdm 5675  cres 5677  Oncon0 6361  Lim wlim 6362  Fun wfun 6534   Fn wfn 6535  cfv 6540  recscrecs 8366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367
This theorem is referenced by:  tfr2b  8392  rdgfun  8412  rdgdmlim  8413  ordtypelem3  9511  ordtypelem4  9512  ordtypelem5  9513  ordtypelem6  9514  ordtypelem7  9515  ordtypelem9  9517
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