Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > orderseqlem | Structured version Visualization version GIF version |
Description: Lemma for poseq 33099 and soseq 33100. 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 6498 | . . . . 5 ⊢ (𝑓 = 𝐺 → (𝑓:𝑥⟶𝐴 ↔ 𝐺:𝑥⟶𝐴)) | |
2 | 1 | rexbidv 3300 | . . . 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 3671 | . . 3 ⊢ (𝐺 ∈ 𝐹 → (𝐺 ∈ 𝐹 ↔ ∃𝑥 ∈ On 𝐺:𝑥⟶𝐴)) |
5 | 4 | ibi 269 | . 2 ⊢ (𝐺 ∈ 𝐹 → ∃𝑥 ∈ On 𝐺:𝑥⟶𝐴) |
6 | frn 6523 | . . . . 5 ⊢ (𝐺:𝑥⟶𝐴 → ran 𝐺 ⊆ 𝐴) | |
7 | unss1 4158 | . . . . 5 ⊢ (ran 𝐺 ⊆ 𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅})) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐺:𝑥⟶𝐴 → (ran 𝐺 ∪ {∅}) ⊆ (𝐴 ∪ {∅})) |
9 | fvrn0 6701 | . . . 4 ⊢ (𝐺‘𝑋) ∈ (ran 𝐺 ∪ {∅}) | |
10 | ssel 3964 | . . . 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 3285 | . 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 1536 ∈ wcel 2113 {cab 2802 ∃wrex 3142 ∪ cun 3937 ⊆ wss 3939 ∅c0 4294 {csn 4570 ran crn 5559 Oncon0 6194 ⟶wf 6354 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 |
This theorem is referenced by: poseq 33099 soseq 33100 |
Copyright terms: Public domain | W3C validator |