Theorem tfr1a 8435

Description: A weak version of tfr1 8438 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 2736	. . . 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 8424	. . 3 ⊢ Fun recs(𝐺)
3		tfr.1	. . . 4 ⊢ 𝐹 = recs(𝐺)
4	3	funeqi 6586	. . 3 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺))
5	2, 4	mpbir 231	. 2 ⊢ Fun 𝐹
6	1	tfrlem16 8434	. . 3 ⊢ Lim dom recs(𝐺)
7	3	dmeqi 5914	. . . 4 ⊢ dom 𝐹 = dom recs(𝐺)
8		limeq 6395	. . . 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 231	. 2 ⊢ Lim dom 𝐹
11	5, 10	pm3.2i 470	1 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539  {cab 2713  ∀wral 3060  ∃wrex 3069  dom cdm 5684   ↾ cres 5686  Oncon0 6383  Lim wlim 6384  Fun wfun 6554   Fn wfn 6555  ‘cfv 6560  recscrecs 8411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412

This theorem is referenced by:  tfr2b  8437  rdgfun  8457  rdgdmlim  8458  ordtypelem3  9561  ordtypelem4  9562  ordtypelem5  9563  ordtypelem6  9564  ordtypelem7  9565  ordtypelem9  9567