Theorem tfrlem6OLD 8393

Description: Obsolete version of tfrlem6 8392 as of 10-Jun-2026. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
tfrlem6OLD	⊢ Rel recs(𝐹)

Distinct variable group:   𝑥,𝑓,𝑦,𝐹

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

Proof of Theorem tfrlem6OLD

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reluni 5795	. . 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 8389	. . . 4 ⊢ (𝑔 ∈ 𝐴 → Fun 𝑔)
4		funrel 6556	. . . 4 ⊢ (Fun 𝑔 → Rel 𝑔)
5	3, 4	syl 18	. . 3 ⊢ (𝑔 ∈ 𝐴 → Rel 𝑔)
6	1, 5	mprgbir 3083	. 2 ⊢ Rel ∪ 𝐴
7	2	recsfval 8391	. . 3 ⊢ recs(𝐹) = ∪ 𝐴
8	7	releqi 5753	. 2 ⊢ (Rel recs(𝐹) ↔ Rel ∪ 𝐴)
9	6, 8	mpbir 234	1 ⊢ Rel recs(𝐹)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086  ∪ cuni 4866   ↾ cres 5652  Rel wrel 5655  Oncon0 6362  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  recscrecs 8381

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-ov 7423  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382

This theorem is used by: (None)