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Theorem orderseqlem 8181
Description: Lemma for poseq 8182 and soseq 8183. The function value of a sequence is either in 𝐴 or null. (Contributed by Scott Fenton, 8-Jun-2011.)
Hypothesis
Ref Expression
orderseqlem.1 𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥𝐴}
Assertion
Ref Expression
orderseqlem (𝐺𝐹 → (𝐺𝑋) ∈ (𝐴 ∪ {∅}))
Distinct variable groups:   𝐴,𝑓,𝑥   𝑓,𝐺,𝑥   𝑥,𝑋
Allowed substitution hints:   𝐹(𝑥,𝑓)   𝑋(𝑓)

Proof of Theorem orderseqlem
StepHypRef Expression
1 feq1 6717 . . . . 5 (𝑓 = 𝐺 → (𝑓:𝑥𝐴𝐺:𝑥𝐴))
21rexbidv 3177 . . . 4 (𝑓 = 𝐺 → (∃𝑥 ∈ On 𝑓:𝑥𝐴 ↔ ∃𝑥 ∈ On 𝐺:𝑥𝐴))
3 orderseqlem.1 . . . 4 𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥𝐴}
42, 3elab2g 3683 . . 3 (𝐺𝐹 → (𝐺𝐹 ↔ ∃𝑥 ∈ On 𝐺:𝑥𝐴))
54ibi 267 . 2 (𝐺𝐹 → ∃𝑥 ∈ On 𝐺:𝑥𝐴)
6 frn 6744 . . . . 5 (𝐺:𝑥𝐴 → ran 𝐺𝐴)
7 unss1 4195 . . . . 5 (ran 𝐺𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
86, 7syl 17 . . . 4 (𝐺:𝑥𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
9 fvrn0 6937 . . . 4 (𝐺𝑋) ∈ (ran 𝐺 ∪ {∅})
10 ssel 3989 . . . 4 ((ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}) → ((𝐺𝑋) ∈ (ran 𝐺 ∪ {∅}) → (𝐺𝑋) ∈ (𝐴 ∪ {∅})))
118, 9, 10mpisyl 21 . . 3 (𝐺:𝑥𝐴 → (𝐺𝑋) ∈ (𝐴 ∪ {∅}))
1211rexlimivw 3149 . 2 (∃𝑥 ∈ On 𝐺:𝑥𝐴 → (𝐺𝑋) ∈ (𝐴 ∪ {∅}))
135, 12syl 17 1 (𝐺𝐹 → (𝐺𝑋) ∈ (𝐴 ∪ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  cun 3961  wss 3963  c0 4339  {csn 4631  ran crn 5690  Oncon0 6386  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  poseq  8182  soseq  8183
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