Theorem tfrlem6 6525

Description: Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)

Hypothesis

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Assertion

Ref	Expression
tfrlem6	⊢ Rel recs(𝐹)

Distinct variable group:   𝑥,𝑓,𝑦,𝐹

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem6

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reluni 4856	. . 3 ⊢ (Rel ∪ 𝐴 ↔ ∀𝑔 ∈ 𝐴 Rel 𝑔)
2		tfrlem.1	. . . . 5 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
3	2	tfrlem4 6522	. . . 4 ⊢ (𝑔 ∈ 𝐴 → Fun 𝑔)
4		funrel 5350	. . . 4 ⊢ (Fun 𝑔 → Rel 𝑔)
5	3, 4	syl 14	. . 3 ⊢ (𝑔 ∈ 𝐴 → Rel 𝑔)
6	1, 5	mprgbir 2591	. 2 ⊢ Rel ∪ 𝐴
7	2	recsfval 6524	. . 3 ⊢ recs(𝐹) = ∪ 𝐴
8	7	releqi 4815	. 2 ⊢ (Rel recs(𝐹) ↔ Rel ∪ 𝐴)
9	6, 8	mpbir 146	1 ⊢ Rel recs(𝐹)

Syntax hints:   ∧ wa 104   = wceq 1398   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512  ∪ cuni 3898  Oncon0 4466   ↾ cres 4733  Rel wrel 4736  Fun wfun 5327   Fn wfn 5328  ‘cfv 5333  recscrecs 6513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-recs 6514

This theorem is referenced by:  tfrlem7  6526