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Theorem orderseqlem 30993
Description: Lemma for poseq 30994 and soseq 30995. 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 5939 . . . . 5 (𝑓 = 𝐺 → (𝑓:𝑥𝐴𝐺:𝑥𝐴))
21rexbidv 3034 . . . 4 (𝑓 = 𝐺 → (∃𝑥 ∈ On 𝑓:𝑥𝐴 ↔ ∃𝑥 ∈ On 𝐺:𝑥𝐴))
3 orderseqlem.1 . . . 4 𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥𝐴}
42, 3elab2g 3322 . . 3 (𝐺𝐹 → (𝐺𝐹 ↔ ∃𝑥 ∈ On 𝐺:𝑥𝐴))
54ibi 255 . 2 (𝐺𝐹 → ∃𝑥 ∈ On 𝐺:𝑥𝐴)
6 frn 5966 . . . . 5 (𝐺:𝑥𝐴 → ran 𝐺𝐴)
7 unss1 3744 . . . . 5 (ran 𝐺𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
86, 7syl 17 . . . 4 (𝐺:𝑥𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
9 fvrn0 6126 . . . 4 (𝐺𝑋) ∈ (ran 𝐺 ∪ {∅})
10 ssel 3562 . . . 4 ((ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}) → ((𝐺𝑋) ∈ (ran 𝐺 ∪ {∅}) → (𝐺𝑋) ∈ (𝐴 ∪ {∅})))
118, 9, 10mpisyl 21 . . 3 (𝐺:𝑥𝐴 → (𝐺𝑋) ∈ (𝐴 ∪ {∅}))
1211rexlimivw 3011 . 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 1475  wcel 1977  {cab 2596  wrex 2897  cun 3538  wss 3540  c0 3874  {csn 4125  ran crn 5039  Oncon0 5640  wf 5800  cfv 5804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812
This theorem is referenced by:  poseq  30994  soseq  30995
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