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Theorem tfrcllemssrecs 6355
Description: Lemma for tfrcl 6367. The union of functions acceptable for tfrcl 6367 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 4493 . . . . . 6 (Ord 𝑋𝑋 ⊆ On)
4 ssrexv 3222 . . . . . 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 3230 . . . 4 (𝜑 → {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))})
71, 6eqsstrid 3203 . . 3 (𝜑𝐴 ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))})
87unissd 3835 . 2 (𝜑 𝐴 {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))})
9 ffn 5367 . . . . . . 7 (𝑓:𝑥𝑆𝑓 Fn 𝑥)
109anim1i 340 . . . . . 6 ((𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) → (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
1110reximi 2574 . . . . 5 (∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) → ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
1211ss2abi 3229 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
1312unissi 3834 . . 3 {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
14 df-recs 6308 . . 3 recs(𝐺) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
1513, 14sseqtrri 3192 . 2 {𝑓 ∣ ∃𝑥 ∈ On (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} ⊆ recs(𝐺)
168, 15sstrdi 3169 1 (𝜑 𝐴 ⊆ recs(𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  {cab 2163  wral 2455  wrex 2456  wss 3131   cuni 3811  Ord word 4364  Oncon0 4365  cres 4630   Fn wfn 5213  wf 5214  cfv 5218  recscrecs 6307
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370  df-f 5222  df-recs 6308
This theorem is referenced by:  tfrcllembfn  6360  tfrcllemubacc  6362  tfrcllemres  6365
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