Theorem tfrcllemssrecs 6242

Description: Lemma for tfrcl 6254. The union of functions acceptable for tfrcl 6254 is a subset of recs. (Contributed by Jim Kingdon, 25-Mar-2022.)

Hypotheses

Ref	Expression
tfrcllemssrecs.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
tfrcllemssrecs.x	⊢ (𝜑 → Ord 𝑋)

Assertion

Ref	Expression
tfrcllemssrecs	⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))

Distinct variable groups:   𝑓,𝐺,𝑥,𝑦   𝑥,𝑋   𝜑,𝑓

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦,𝑓)   𝑆(𝑥,𝑦,𝑓)   𝑋(𝑦,𝑓)

Proof of Theorem tfrcllemssrecs

Step	Hyp	Ref	Expression
1		tfrcllemssrecs.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
2		tfrcllemssrecs.x	. . . . . 6 ⊢ (𝜑 → Ord 𝑋)
3		ordsson 4403	. . . . . 6 ⊢ (Ord 𝑋 → 𝑋 ⊆ On)
4		ssrexv 3157	. . . . . 6 ⊢ (𝑋 ⊆ On → (∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) → ∃𝑥 ∈ On (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))))
5	2, 3, 4	3syl 17	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) → ∃𝑥 ∈ On (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))))
6	5	ss2abdv 3165	. . . 4 ⊢ (𝜑 → {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))})
7	1, 6	eqsstrid 3138	. . 3 ⊢ (𝜑 → 𝐴 ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))})
8	7	unissd 3755	. 2 ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))})
9		ffn 5267	. . . . . . 7 ⊢ (𝑓:𝑥⟶𝑆 → 𝑓 Fn 𝑥)
10	9	anim1i 338	. . . . . 6 ⊢ ((𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) → (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))
11	10	reximi 2527	. . . . 5 ⊢ (∃𝑥 ∈ On (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) → ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))
12	11	ss2abi 3164	. . . 4 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
13	12	unissi 3754	. . 3 ⊢ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} ⊆ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
14		df-recs 6195	. . 3 ⊢ recs(𝐺) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
15	13, 14	sseqtrri 3127	. 2 ⊢ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} ⊆ recs(𝐺)
16	8, 15	sstrdi 3104	1 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331  {cab 2123  ∀wral 2414  ∃wrex 2415   ⊆ wss 3066  ∪ cuni 3731  Ord word 4279  Oncon0 4280   ↾ cres 4536   Fn wfn 5113  ⟶wf 5114  ‘cfv 5118  recscrecs 6194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285  df-f 5122  df-recs 6195

This theorem is referenced by:  tfrcllembfn  6247  tfrcllemubacc  6249  tfrcllemres  6252