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Theorem tfr1a 7757
 Description: A weak version of tfr1 7760 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 2826 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem7 7746 . . 3 Fun recs(𝐺)
3 tfr.1 . . . 4 𝐹 = recs(𝐺)
43funeqi 6145 . . 3 (Fun 𝐹 ↔ Fun recs(𝐺))
52, 4mpbir 223 . 2 Fun 𝐹
61tfrlem16 7756 . . 3 Lim dom recs(𝐺)
73dmeqi 5558 . . . 4 dom 𝐹 = dom recs(𝐺)
8 limeq 5976 . . . 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 223 . 2 Lim dom 𝐹
115, 10pm3.2i 464 1 (Fun 𝐹 ∧ Lim dom 𝐹)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1658  {cab 2812  ∀wral 3118  ∃wrex 3119  dom cdm 5343   ↾ cres 5345  Oncon0 5964  Lim wlim 5965  Fun wfun 6118   Fn wfn 6119  ‘cfv 6124  recscrecs 7734 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-wrecs 7673  df-recs 7735 This theorem is referenced by:  tfr2b  7759  rdgfun  7779  rdgdmlim  7780  ordtypelem3  8695  ordtypelem4  8696  ordtypelem5  8697  ordtypelem6  8698  ordtypelem7  8699  ordtypelem9  8701
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