Theorem orderseqlem 8162

Description: Lemma for poseq 8163 and soseq 8164. 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:𝑥⟶𝐴}

Assertion

Ref	Expression
orderseqlem	⊢ (𝐺 ∈ 𝐹 → (𝐺‘𝑋) ∈ (𝐴 ∪ {∅}))

Distinct variable groups:   𝐴,𝑓,𝑥   𝑓,𝐺,𝑥   𝑥,𝑋

Allowed substitution hints:   𝐹(𝑥,𝑓)   𝑋(𝑓)

Proof of Theorem orderseqlem

Step	Hyp	Ref	Expression
1		feq1 6704	. . . . 5 ⊢ (𝑓 = 𝐺 → (𝑓:𝑥⟶𝐴 ↔ 𝐺:𝑥⟶𝐴))
2	1	rexbidv 3168	. . . 4 ⊢ (𝑓 = 𝐺 → (∃𝑥 ∈ On 𝑓:𝑥⟶𝐴 ↔ ∃𝑥 ∈ On 𝐺:𝑥⟶𝐴))
3		orderseqlem.1	. . . 4 ⊢ 𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶𝐴}
4	2, 3	elab2g 3666	. . 3 ⊢ (𝐺 ∈ 𝐹 → (𝐺 ∈ 𝐹 ↔ ∃𝑥 ∈ On 𝐺:𝑥⟶𝐴))
5	4	ibi 266	. 2 ⊢ (𝐺 ∈ 𝐹 → ∃𝑥 ∈ On 𝐺:𝑥⟶𝐴)
6		frn 6730	. . . . 5 ⊢ (𝐺:𝑥⟶𝐴 → ran 𝐺 ⊆ 𝐴)
7		unss1 4177	. . . . 5 ⊢ (ran 𝐺 ⊆ 𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
8	6, 7	syl 17	. . . 4 ⊢ (𝐺:𝑥⟶𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
9		fvrn0 6926	. . . 4 ⊢ (𝐺‘𝑋) ∈ (ran 𝐺 ∪ {∅})
10		ssel 3970	. . . 4 ⊢ ((ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}) → ((𝐺‘𝑋) ∈ (ran 𝐺 ∪ {∅}) → (𝐺‘𝑋) ∈ (𝐴 ∪ {∅})))
11	8, 9, 10	mpisyl 21	. . 3 ⊢ (𝐺:𝑥⟶𝐴 → (𝐺‘𝑋) ∈ (𝐴 ∪ {∅}))
12	11	rexlimivw 3140	. 2 ⊢ (∃𝑥 ∈ On 𝐺:𝑥⟶𝐴 → (𝐺‘𝑋) ∈ (𝐴 ∪ {∅}))
13	5, 12	syl 17	1 ⊢ (𝐺 ∈ 𝐹 → (𝐺‘𝑋) ∈ (𝐴 ∪ {∅}))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {cab 2702  ∃wrex 3059   ∪ cun 3942   ⊆ wss 3944  ∅c0 4322  {csn 4630  ran crn 5679  Oncon0 6371  ⟶wf 6545  ‘cfv 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557

This theorem is referenced by:  poseq  8163  soseq  8164