Theorem orderseqlem 33801

Description: Lemma for poseq 33802 and soseq 33803. 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: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 6581	. . . . 5 ⊢ (𝑓 = 𝐺 → (𝑓:𝑥⟶𝐴 ↔ 𝐺:𝑥⟶𝐴))
2	1	rexbidv 3226	. . . 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 3611	. . 3 ⊢ (𝐺 ∈ 𝐹 → (𝐺 ∈ 𝐹 ↔ ∃𝑥 ∈ On 𝐺:𝑥⟶𝐴))
5	4	ibi 266	. 2 ⊢ (𝐺 ∈ 𝐹 → ∃𝑥 ∈ On 𝐺:𝑥⟶𝐴)
6		frn 6607	. . . . 5 ⊢ (𝐺:𝑥⟶𝐴 → ran 𝐺 ⊆ 𝐴)
7		unss1 4113	. . . . 5 ⊢ (ran 𝐺 ⊆ 𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
8	6, 7	syl 17	. . . 4 ⊢ (𝐺:𝑥⟶𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅}))
9		fvrn0 6802	. . . 4 ⊢ (𝐺‘𝑋) ∈ (ran 𝐺 ∪ {∅})
10		ssel 3914	. . . 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 3211	. 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 1539   ∈ wcel 2106  {cab 2715  ∃wrex 3065   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  {csn 4561  ran crn 5590  Oncon0 6266  ⟶wf 6429  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441

This theorem is referenced by:  poseq  33802  soseq  33803