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Theorem tfrcllemssrecs 6099
Description: Lemma for tfrcl 6111. The union of functions acceptable for tfrcl 6111 is a subset of recs. (Contributed by Jim Kingdon, 25-Mar-2022.)
Hypotheses
Ref Expression
tfrcllemssrecs.1 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
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
StepHypRef Expression
1 tfrcllemssrecs.1 . . . 4 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
2 tfrcllemssrecs.x . . . . . 6 (𝜑 → Ord 𝑋)
3 ordsson 4299 . . . . . 6 (Ord 𝑋𝑋 ⊆ On)
4 ssrexv 3084 . . . . . 6 (𝑋 ⊆ On → (∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) → ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))))
52, 3, 43syl 17 . . . . 5 (𝜑 → (∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) → ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))))
65ss2abdv 3092 . . . 4 (𝜑 → {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))})
71, 6syl5eqss 3068 . . 3 (𝜑𝐴 ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))})
87unissd 3672 . 2 (𝜑 𝐴 {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))})
9 ffn 5147 . . . . . . 7 (𝑓:𝑥𝑆𝑓 Fn 𝑥)
109anim1i 333 . . . . . 6 ((𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) → (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
1110reximi 2470 . . . . 5 (∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) → ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
1211ss2abi 3091 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
1312unissi 3671 . . 3 {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
14 df-recs 6052 . . 3 recs(𝐺) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
1513, 14sseqtr4i 3057 . 2 {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} ⊆ recs(𝐺)
168, 15syl6ss 3035 1 (𝜑 𝐴 ⊆ recs(𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  {cab 2074  wral 2359  wrex 2360  wss 2997   cuni 3648  Ord word 4180  Oncon0 4181  cres 4430   Fn wfn 4997  wf 4998  cfv 5002  recscrecs 6051
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3003  df-ss 3010  df-uni 3649  df-tr 3929  df-iord 4184  df-on 4186  df-f 5006  df-recs 6052
This theorem is referenced by:  tfrcllembfn  6104  tfrcllemubacc  6106  tfrcllemres  6109
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