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Theorem orderseqlem 30835
Description: Lemma for poseq 30836 and soseq 30837. The function value of a sequene 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 5824 . . . . 5 (𝑓 = 𝐺 → (𝑓:𝑥𝐴𝐺:𝑥𝐴))
21rexbidv 2938 . . . 4 (𝑓 = 𝐺 → (∃𝑥 ∈ On 𝑓:𝑥𝐴 ↔ ∃𝑥 ∈ On 𝐺:𝑥𝐴))
3 orderseqlem.1 . . . 4 𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥𝐴}
42, 3elab2g 3226 . . 3 (𝐺𝐹 → (𝐺𝐹 ↔ ∃𝑥 ∈ On 𝐺:𝑥𝐴))
54ibi 254 . 2 (𝐺𝐹 → ∃𝑥 ∈ On 𝐺:𝑥𝐴)
6 frn 5851 . . . . 5 (𝐺:𝑥𝐴 → ran 𝐺𝐴)
7 unss1 3648 . . . . 5 (ran 𝐺𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
86, 7syl 17 . . . 4 (𝐺:𝑥𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
9 fvrn0 6010 . . . 4 (𝐺𝑋) ∈ (ran 𝐺 ∪ {∅})
10 ssel 3466 . . . 4 ((ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}) → ((𝐺𝑋) ∈ (ran 𝐺 ∪ {∅}) → (𝐺𝑋) ∈ (𝐴 ∪ {∅})))
118, 9, 10mpisyl 21 . . 3 (𝐺:𝑥𝐴 → (𝐺𝑋) ∈ (𝐴 ∪ {∅}))
1211rexlimivw 2915 . 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 1474  wcel 1938  {cab 2500  wrex 2801  cun 3442  wss 3444  c0 3777  {csn 4028  ran crn 4933  Oncon0 5530  wf 5685  cfv 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-fv 5697
This theorem is referenced by:  poseq  30836  soseq  30837
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