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Theorem tfr1onlemssrecs 6039
Description: Lemma for tfr1on 6050. The union of functions acceptable for tfr1on 6050 is a subset of recs. (Contributed by Jim Kingdon, 15-Mar-2022.)
Hypotheses
Ref Expression
tfr1onlemssrecs.1 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
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
StepHypRef Expression
1 tfr1onlemssrecs.1 . . . 4 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
2 tfr1onlemssrecs.x . . . . . 6 (𝜑 → Ord 𝑋)
3 ordsson 4275 . . . . . 6 (Ord 𝑋𝑋 ⊆ On)
4 ssrexv 3072 . . . . . 6 (𝑋 ⊆ On → (∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) → ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))))
52, 3, 43syl 17 . . . . 5 (𝜑 → (∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) → ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))))
65ss2abdv 3080 . . . 4 (𝜑 → {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))})
71, 6syl5eqss 3056 . . 3 (𝜑𝐴 ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))})
87unissd 3654 . 2 (𝜑 𝐴 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))})
9 df-recs 6005 . 2 recs(𝐺) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
108, 9syl6sseqr 3059 1 (𝜑 𝐴 ⊆ recs(𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  {cab 2071  wral 2355  wrex 2356  wss 2986   cuni 3630  Ord word 4156  Oncon0 4157  cres 4406   Fn wfn 4967  cfv 4972  recscrecs 6004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-in 2992  df-ss 2999  df-uni 3631  df-tr 3905  df-iord 4160  df-on 4162  df-recs 6005
This theorem is referenced by:  tfr1onlembfn  6044  tfr1onlemubacc  6046  tfr1onlemres  6049
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