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Mirrors > Home > MPE Home > Th. List > Mathboxes > orderseqlem | Structured version Visualization version GIF version |
Description: Lemma for poseq 33208 and soseq 33209. The function value of a sequene 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 6468 | . . . . 5 ⊢ (𝑓 = 𝐺 → (𝑓:𝑥⟶𝐴 ↔ 𝐺:𝑥⟶𝐴)) | |
2 | 1 | rexbidv 3256 | . . . 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 3616 | . . 3 ⊢ (𝐺 ∈ 𝐹 → (𝐺 ∈ 𝐹 ↔ ∃𝑥 ∈ On 𝐺:𝑥⟶𝐴)) |
5 | 4 | ibi 270 | . 2 ⊢ (𝐺 ∈ 𝐹 → ∃𝑥 ∈ On 𝐺:𝑥⟶𝐴) |
6 | frn 6493 | . . . . 5 ⊢ (𝐺:𝑥⟶𝐴 → ran 𝐺 ⊆ 𝐴) | |
7 | unss1 4106 | . . . . 5 ⊢ (ran 𝐺 ⊆ 𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅})) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐺:𝑥⟶𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅})) |
9 | fvrn0 6673 | . . . 4 ⊢ (𝐺‘𝑋) ∈ (ran 𝐺 ∪ {∅}) | |
10 | ssel 3908 | . . . 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 3241 | . 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 1538 ∈ wcel 2111 {cab 2776 ∃wrex 3107 ∪ cun 3879 ⊆ wss 3881 ∅c0 4243 {csn 4525 ran crn 5520 Oncon0 6159 ⟶wf 6320 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 |
This theorem is referenced by: poseq 33208 soseq 33209 |
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