Theorem tfrlem6OLD 8375

Description: Obsolete version of tfrlem6 8374 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 5803	. . 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 8371	. . . 4 ⊢ (𝑔 ∈ 𝐴 → Fun 𝑔)
4		funrel 6554	. . . 4 ⊢ (Fun 𝑔 → Rel 𝑔)
5	3, 4	syl 18	. . 3 ⊢ (𝑔 ∈ 𝐴 → Rel 𝑔)
6	1, 5	mprgbir 3085	. 2 ⊢ Rel ∪ 𝐴
7	2	recsfval 8373	. . 3 ⊢ recs(𝐹) = ∪ 𝐴
8	7	releqi 5762	. 2 ⊢ (Rel recs(𝐹) ↔ Rel ∪ 𝐴)
9	6, 8	mpbir 234	1 ⊢ Rel recs(𝐹)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  ∀wral 3078  ∃wrex 3088  ∪ cuni 4870   ↾ cres 5661  Rel wrel 5664  Oncon0 6361  Fun wfun 6531   Fn wfn 6532  ‘cfv 6537  recscrecs 8363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-ov 7420  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364

This theorem is used by: (None)