Theorem tfr1onlemssrecs 6394

Description: Lemma for tfr1on 6405. The union of functions acceptable for tfr1on 6405 is a subset of recs. (Contributed by Jim Kingdon, 15-Mar-2022.)

Hypotheses

Ref	Expression
tfr1onlemssrecs.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
tfr1onlemssrecs.x	⊢ (𝜑 → Ord 𝑋)

Assertion

Ref	Expression
tfr1onlemssrecs	⊢ (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)   𝑋(𝑦,𝑓)

Proof of Theorem tfr1onlemssrecs

Step	Hyp	Ref	Expression
1		tfr1onlemssrecs.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
2		tfr1onlemssrecs.x	. . . . . 6 ⊢ (𝜑 → Ord 𝑋)
3		ordsson 4525	. . . . . 6 ⊢ (Ord 𝑋 → 𝑋 ⊆ On)
4		ssrexv 3245	. . . . . 6 ⊢ (𝑋 ⊆ On → (∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) → ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))))
5	2, 3, 4	3syl 17	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) → ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))))
6	5	ss2abdv 3253	. . . 4 ⊢ (𝜑 → {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))})
7	1, 6	eqsstrid 3226	. . 3 ⊢ (𝜑 → 𝐴 ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))})
8	7	unissd 3860	. 2 ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))})
9		df-recs 6360	. 2 ⊢ recs(𝐺) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
10	8, 9	sseqtrrdi 3229	1 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  {cab 2179  ∀wral 2472  ∃wrex 2473   ⊆ wss 3154  ∪ cuni 3836  Ord word 4394  Oncon0 4395   ↾ cres 4662   Fn wfn 5250  ‘cfv 5255  recscrecs 6359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3160  df-ss 3167  df-uni 3837  df-tr 4129  df-iord 4398  df-on 4400  df-recs 6360

This theorem is referenced by:  tfr1onlembfn  6399  tfr1onlemubacc  6401  tfr1onlemres  6404