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Theorem tfr1a 8381
Description: A weak version of tfr1 8384 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 2733 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem7 8370 . . 3 Fun recs(𝐺)
3 tfr.1 . . . 4 𝐹 = recs(𝐺)
43funeqi 6561 . . 3 (Fun 𝐹 ↔ Fun recs(𝐺))
52, 4mpbir 230 . 2 Fun 𝐹
61tfrlem16 8380 . . 3 Lim dom recs(𝐺)
73dmeqi 5899 . . . 4 dom 𝐹 = dom recs(𝐺)
8 limeq 6368 . . . 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 472 1 (Fun 𝐹 ∧ Lim dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  {cab 2710  wral 3062  wrex 3071  dom cdm 5672  cres 5674  Oncon0 6356  Lim wlim 6357  Fun wfun 6529   Fn wfn 6530  cfv 6535  recscrecs 8357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358
This theorem is referenced by:  tfr2b  8383  rdgfun  8403  rdgdmlim  8404  ordtypelem3  9502  ordtypelem4  9503  ordtypelem5  9504  ordtypelem6  9505  ordtypelem7  9506  ordtypelem9  9508
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