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Theorem tfr1a 8008
Description: A weak version of tfr1 8011 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 2820 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem7 7997 . . 3 Fun recs(𝐺)
3 tfr.1 . . . 4 𝐹 = recs(𝐺)
43funeqi 6352 . . 3 (Fun 𝐹 ↔ Fun recs(𝐺))
52, 4mpbir 233 . 2 Fun 𝐹
61tfrlem16 8007 . . 3 Lim dom recs(𝐺)
73dmeqi 5749 . . . 4 dom 𝐹 = dom recs(𝐺)
8 limeq 6179 . . . 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 233 . 2 Lim dom 𝐹
115, 10pm3.2i 473 1 (Fun 𝐹 ∧ Lim dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1537  {cab 2798  wral 3125  wrex 3126  dom cdm 5531  cres 5533  Oncon0 6167  Lim wlim 6168  Fun wfun 6325   Fn wfn 6326  cfv 6331  recscrecs 7985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-wrecs 7925  df-recs 7986
This theorem is referenced by:  tfr2b  8010  rdgfun  8030  rdgdmlim  8031  ordtypelem3  8962  ordtypelem4  8963  ordtypelem5  8964  ordtypelem6  8965  ordtypelem7  8966  ordtypelem9  8968
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