Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > snelmap | Structured version Visualization version GIF version |
Description: Membership of the element in the range of a constant map. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
snelmap.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
snelmap.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
snelmap.n | ⊢ (𝜑 → 𝐴 ≠ ∅) |
snelmap.e | ⊢ (𝜑 → (𝐴 × {𝑥}) ∈ (𝐵 ↑m 𝐴)) |
Ref | Expression |
---|---|
snelmap | ⊢ (𝜑 → 𝑥 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snelmap.n | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
2 | n0 4307 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴) | |
3 | 1, 2 | sylib 219 | . 2 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ 𝐴) |
4 | vex 3495 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | 4 | fvconst2 6958 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ((𝐴 × {𝑥})‘𝑦) = 𝑥) |
6 | 5 | eqcomd 2824 | . . . . . 6 ⊢ (𝑦 ∈ 𝐴 → 𝑥 = ((𝐴 × {𝑥})‘𝑦)) |
7 | 6 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑥 = ((𝐴 × {𝑥})‘𝑦)) |
8 | snelmap.e | . . . . . . . 8 ⊢ (𝜑 → (𝐴 × {𝑥}) ∈ (𝐵 ↑m 𝐴)) | |
9 | snelmap.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
10 | snelmap.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
11 | elmapg 8408 | . . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ((𝐴 × {𝑥}) ∈ (𝐵 ↑m 𝐴) ↔ (𝐴 × {𝑥}):𝐴⟶𝐵)) | |
12 | 9, 10, 11 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → ((𝐴 × {𝑥}) ∈ (𝐵 ↑m 𝐴) ↔ (𝐴 × {𝑥}):𝐴⟶𝐵)) |
13 | 8, 12 | mpbid 233 | . . . . . . 7 ⊢ (𝜑 → (𝐴 × {𝑥}):𝐴⟶𝐵) |
14 | 13 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐴 × {𝑥}):𝐴⟶𝐵) |
15 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
16 | 14, 15 | ffvelrnd 6844 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐴 × {𝑥})‘𝑦) ∈ 𝐵) |
17 | 7, 16 | eqeltrd 2910 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
18 | 17 | ex 413 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
19 | 18 | exlimdv 1925 | . 2 ⊢ (𝜑 → (∃𝑦 𝑦 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
20 | 3, 19 | mpd 15 | 1 ⊢ (𝜑 → 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 {csn 4557 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 |
This theorem is referenced by: mapssbi 41352 |
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