Theorem tfr1a 7435

Description: A weak version of tfr1 7438 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 2621	. . . 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 7424	. . 3 ⊢ Fun recs(𝐺)
3		tfr.1	. . . 4 ⊢ 𝐹 = recs(𝐺)
4	3	funeqi 5868	. . 3 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺))
5	2, 4	mpbir 221	. 2 ⊢ Fun 𝐹
6	1	tfrlem16 7434	. . 3 ⊢ Lim dom recs(𝐺)
7	3	dmeqi 5285	. . . 4 ⊢ dom 𝐹 = dom recs(𝐺)
8		limeq 5694	. . . 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 221	. 2 ⊢ Lim dom 𝐹
11	5, 10	pm3.2i 471	1 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480  {cab 2607  ∀wral 2907  ∃wrex 2908  dom cdm 5074   ↾ cres 5076  Oncon0 5682  Lim wlim 5683  Fun wfun 5841   Fn wfn 5842  ‘cfv 5847  recscrecs 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-wrecs 7352  df-recs 7413

This theorem is referenced by:  tfr2b  7437  rdgfun  7457  rdgdmlim  7458  ordtypelem3  8369  ordtypelem4  8370  ordtypelem5  8371  ordtypelem6  8372  ordtypelem7  8373  ordtypelem9  8375