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Theorem tfr1a 8408
Description: A weak version of tfr1 8411 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 2728 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem7 8397 . . 3 Fun recs(𝐺)
3 tfr.1 . . . 4 𝐹 = recs(𝐺)
43funeqi 6568 . . 3 (Fun 𝐹 ↔ Fun recs(𝐺))
52, 4mpbir 230 . 2 Fun 𝐹
61tfrlem16 8407 . . 3 Lim dom recs(𝐺)
73dmeqi 5901 . . . 4 dom 𝐹 = dom recs(𝐺)
8 limeq 6375 . . . 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 470 1 (Fun 𝐹 ∧ Lim dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  {cab 2705  wral 3057  wrex 3066  dom cdm 5672  cres 5674  Oncon0 6363  Lim wlim 6364  Fun wfun 6536   Fn wfn 6537  cfv 6542  recscrecs 8384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385
This theorem is referenced by:  tfr2b  8410  rdgfun  8430  rdgdmlim  8431  ordtypelem3  9537  ordtypelem4  9538  ordtypelem5  9539  ordtypelem6  9540  ordtypelem7  9541  ordtypelem9  9543
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