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Mirrors > Home > MPE Home > Th. List > orderseqlem | Structured version Visualization version GIF version |
Description: Lemma for poseq 8182 and soseq 8183. The function value of a sequence is either in 𝐴 or null. (Contributed by Scott Fenton, 8-Jun-2011.) |
Ref | Expression |
---|---|
orderseqlem.1 | ⊢ 𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶𝐴} |
Ref | Expression |
---|---|
orderseqlem | ⊢ (𝐺 ∈ 𝐹 → (𝐺‘𝑋) ∈ (𝐴 ∪ {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq1 6717 | . . . . 5 ⊢ (𝑓 = 𝐺 → (𝑓:𝑥⟶𝐴 ↔ 𝐺:𝑥⟶𝐴)) | |
2 | 1 | rexbidv 3177 | . . . 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 3683 | . . 3 ⊢ (𝐺 ∈ 𝐹 → (𝐺 ∈ 𝐹 ↔ ∃𝑥 ∈ On 𝐺:𝑥⟶𝐴)) |
5 | 4 | ibi 267 | . 2 ⊢ (𝐺 ∈ 𝐹 → ∃𝑥 ∈ On 𝐺:𝑥⟶𝐴) |
6 | frn 6744 | . . . . 5 ⊢ (𝐺:𝑥⟶𝐴 → ran 𝐺 ⊆ 𝐴) | |
7 | unss1 4195 | . . . . 5 ⊢ (ran 𝐺 ⊆ 𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅})) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐺:𝑥⟶𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅})) |
9 | fvrn0 6937 | . . . 4 ⊢ (𝐺‘𝑋) ∈ (ran 𝐺 ∪ {∅}) | |
10 | ssel 3989 | . . . 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 3149 | . 2 ⊢ (∃𝑥 ∈ On 𝐺:𝑥⟶𝐴 → (𝐺‘𝑋) ∈ (𝐴 ∪ {∅})) |
13 | 5, 12 | syl 17 | 1 ⊢ (𝐺 ∈ 𝐹 → (𝐺‘𝑋) ∈ (𝐴 ∪ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {cab 2712 ∃wrex 3068 ∪ cun 3961 ⊆ wss 3963 ∅c0 4339 {csn 4631 ran crn 5690 Oncon0 6386 ⟶wf 6559 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 |
This theorem is referenced by: poseq 8182 soseq 8183 |
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