Theorem tfr1a 8381

Description: A weak version of tfr1 8384 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 2733	. . . 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 8370	. . 3 ⊢ Fun recs(𝐺)
3		tfr.1	. . . 4 ⊢ 𝐹 = recs(𝐺)
4	3	funeqi 6561	. . 3 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺))
5	2, 4	mpbir 230	. 2 ⊢ Fun 𝐹
6	1	tfrlem16 8380	. . 3 ⊢ Lim dom recs(𝐺)
7	3	dmeqi 5899	. . . 4 ⊢ dom 𝐹 = dom recs(𝐺)
8		limeq 6368	. . . 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 472	1 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542  {cab 2710  ∀wral 3062  ∃wrex 3071  dom cdm 5672   ↾ cres 5674  Oncon0 6356  Lim wlim 6357  Fun wfun 6529   Fn wfn 6530  ‘cfv 6535  recscrecs 8357

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423  ax-un 7712

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  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 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358

This theorem is referenced by:  tfr2b  8383  rdgfun  8403  rdgdmlim  8404  ordtypelem3  9502  ordtypelem4  9503  ordtypelem5  9504  ordtypelem6  9505  ordtypelem7  9506  ordtypelem9  9508