Theorem tfr1a 7490

Description: A weak version of tfr1 7493 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 2622	. . . 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 7479	. . 3 ⊢ Fun recs(𝐺)
3		tfr.1	. . . 4 ⊢ 𝐹 = recs(𝐺)
4	3	funeqi 5909	. . 3 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺))
5	2, 4	mpbir 221	. 2 ⊢ Fun 𝐹
6	1	tfrlem16 7489	. . 3 ⊢ Lim dom recs(𝐺)
7	3	dmeqi 5325	. . . 4 ⊢ dom 𝐹 = dom recs(𝐺)
8		limeq 5735	. . . 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 1483  {cab 2608  ∀wral 2912  ∃wrex 2913  dom cdm 5114   ↾ cres 5116  Oncon0 5723  Lim wlim 5724  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888  recscrecs 7467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  tfr2b  7492  rdgfun  7512  rdgdmlim  7513  ordtypelem3  8425  ordtypelem4  8426  ordtypelem5  8427  ordtypelem6  8428  ordtypelem7  8429  ordtypelem9  8431