Theorem tfrlem6OLD 8368

Description: Obsolete version of tfrlem6 8367 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 5805	. . 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 8364	. . . 4 ⊢ (𝑔 ∈ 𝐴 → Fun 𝑔)
4		funrel 6553	. . . 4 ⊢ (Fun 𝑔 → Rel 𝑔)
5	3, 4	syl 18	. . 3 ⊢ (𝑔 ∈ 𝐴 → Rel 𝑔)
6	1, 5	mprgbir 3084	. 2 ⊢ Rel ∪ 𝐴
7	2	recsfval 8366	. . 3 ⊢ recs(𝐹) = ∪ 𝐴
8	7	releqi 5764	. 2 ⊢ (Rel recs(𝐹) ↔ Rel ∪ 𝐴)
9	6, 8	mpbir 234	1 ⊢ Rel recs(𝐹)

Syntax hints:   ∧ wa 400   = wceq 1568   ∈ wcel 2141  {cab 2739  ∀wral 3077  ∃wrex 3087  ∪ cuni 4871   ↾ cres 5663  Rel wrel 5666  Oncon0 6360  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  recscrecs 8356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-ov 7413  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357

This theorem is referenced by: (None)