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Theorem orderseqlem 32703
 Description: Lemma for poseq 32704 and soseq 32705. 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 6363 . . . . 5 (𝑓 = 𝐺 → (𝑓:𝑥𝐴𝐺:𝑥𝐴))
21rexbidv 3260 . . . 4 (𝑓 = 𝐺 → (∃𝑥 ∈ On 𝑓:𝑥𝐴 ↔ ∃𝑥 ∈ On 𝐺:𝑥𝐴))
3 orderseqlem.1 . . . 4 𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥𝐴}
42, 3elab2g 3607 . . 3 (𝐺𝐹 → (𝐺𝐹 ↔ ∃𝑥 ∈ On 𝐺:𝑥𝐴))
54ibi 268 . 2 (𝐺𝐹 → ∃𝑥 ∈ On 𝐺:𝑥𝐴)
6 frn 6388 . . . . 5 (𝐺:𝑥𝐴 → ran 𝐺𝐴)
7 unss1 4076 . . . . 5 (ran 𝐺𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
86, 7syl 17 . . . 4 (𝐺:𝑥𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
9 fvrn0 6566 . . . 4 (𝐺𝑋) ∈ (ran 𝐺 ∪ {∅})
10 ssel 3883 . . . 4 ((ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}) → ((𝐺𝑋) ∈ (ran 𝐺 ∪ {∅}) → (𝐺𝑋) ∈ (𝐴 ∪ {∅})))
118, 9, 10mpisyl 21 . . 3 (𝐺:𝑥𝐴 → (𝐺𝑋) ∈ (𝐴 ∪ {∅}))
1211rexlimivw 3245 . 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 1522   ∈ wcel 2081  {cab 2775  ∃wrex 3106   ∪ cun 3857   ⊆ wss 3859  ∅c0 4211  {csn 4472  ran crn 5444  Oncon0 6066  ⟶wf 6221  ‘cfv 6225 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233 This theorem is referenced by:  poseq  32704  soseq  32705
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