Theorem tfr1a 8225

Description: A weak version of tfr1 8228 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.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem7 8214	. . 3 ⊢ Fun recs(𝐺)
3		tfr.1	. . . 4 ⊢ 𝐹 = recs(𝐺)
4	3	funeqi 6455	. . 3 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺))
5	2, 4	mpbir 230	. 2 ⊢ Fun 𝐹
6	1	tfrlem16 8224	. . 3 ⊢ Lim dom recs(𝐺)
7	3	dmeqi 5813	. . . 4 ⊢ dom 𝐹 = dom recs(𝐺)
8		limeq 6278	. . . 4 ⊢ (dom 𝐹 = dom recs(𝐺) → (Lim dom 𝐹 ↔ Lim dom recs(𝐺)))
9	7, 8	ax-mp 5	. . 3 ⊢ (Lim dom 𝐹 ↔ Lim dom recs(𝐺))
10	6, 9	mpbir 230	. 2 ⊢ Lim dom 𝐹
11	5, 10	pm3.2i 471	1 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539  {cab 2715  ∀wral 3064  ∃wrex 3065  dom cdm 5589   ↾ cres 5591  Oncon0 6266  Lim wlim 6267  Fun wfun 6427   Fn wfn 6428  ‘cfv 6433  recscrecs 8201

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202

This theorem is referenced by:  tfr2b  8227  rdgfun  8247  rdgdmlim  8248  ordtypelem3  9279  ordtypelem4  9280  ordtypelem5  9281  ordtypelem6  9282  ordtypelem7  9283  ordtypelem9  9285