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Theorem orderseqlem 31723
Description: Lemma for poseq 31724 and soseq 31725. 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 6013 . . . . 5 (𝑓 = 𝐺 → (𝑓:𝑥𝐴𝐺:𝑥𝐴))
21rexbidv 3048 . . . 4 (𝑓 = 𝐺 → (∃𝑥 ∈ On 𝑓:𝑥𝐴 ↔ ∃𝑥 ∈ On 𝐺:𝑥𝐴))
3 orderseqlem.1 . . . 4 𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥𝐴}
42, 3elab2g 3347 . . 3 (𝐺𝐹 → (𝐺𝐹 ↔ ∃𝑥 ∈ On 𝐺:𝑥𝐴))
54ibi 256 . 2 (𝐺𝐹 → ∃𝑥 ∈ On 𝐺:𝑥𝐴)
6 frn 6040 . . . . 5 (𝐺:𝑥𝐴 → ran 𝐺𝐴)
7 unss1 3774 . . . . 5 (ran 𝐺𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
86, 7syl 17 . . . 4 (𝐺:𝑥𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
9 fvrn0 6203 . . . 4 (𝐺𝑋) ∈ (ran 𝐺 ∪ {∅})
10 ssel 3589 . . . 4 ((ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}) → ((𝐺𝑋) ∈ (ran 𝐺 ∪ {∅}) → (𝐺𝑋) ∈ (𝐴 ∪ {∅})))
118, 9, 10mpisyl 21 . . 3 (𝐺:𝑥𝐴 → (𝐺𝑋) ∈ (𝐴 ∪ {∅}))
1211rexlimivw 3025 . 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 1481  wcel 1988  {cab 2606  wrex 2910  cun 3565  wss 3567  c0 3907  {csn 4168  ran crn 5105  Oncon0 5711  wf 5872  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884
This theorem is referenced by:  poseq  31724  soseq  31725
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