Theorem tfrlem6 8409

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 5824	. . 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 8406	. . . 4 ⊢ (𝑔 ∈ 𝐴 → Fun 𝑔)
4		funrel 6575	. . . 4 ⊢ (Fun 𝑔 → Rel 𝑔)
5	3, 4	syl 17	. . 3 ⊢ (𝑔 ∈ 𝐴 → Rel 𝑔)
6	1, 5	mprgbir 3065	. 2 ⊢ Rel ∪ 𝐴
7	2	recsfval 8408	. . 3 ⊢ recs(𝐹) = ∪ 𝐴
8	7	releqi 5783	. 2 ⊢ (Rel recs(𝐹) ↔ Rel ∪ 𝐴)
9	6, 8	mpbir 230	1 ⊢ Rel recs(𝐹)

Syntax hints:   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2705  ∀wral 3058  ∃wrex 3067  ∪ cuni 4912   ↾ cres 5684  Rel wrel 5687  Oncon0 6374  Fun wfun 6547   Fn wfn 6548  ‘cfv 6553  recscrecs 8397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561  df-ov 7429  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398

This theorem is referenced by:  tfrlem7  8410  tfrlem11  8415  tfrlem15  8419  tfrlem16  8420