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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 4305 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴) | |
3 | 1, 2 | sylib 217 | . 2 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ 𝐴) |
4 | vex 3448 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | 4 | fvconst2 7150 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ((𝐴 × {𝑥})‘𝑦) = 𝑥) |
6 | 5 | eqcomd 2744 | . . . . . 6 ⊢ (𝑦 ∈ 𝐴 → 𝑥 = ((𝐴 × {𝑥})‘𝑦)) |
7 | 6 | adantl 483 | . . . . 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 8737 | . . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ((𝐴 × {𝑥}) ∈ (𝐵 ↑m 𝐴) ↔ (𝐴 × {𝑥}):𝐴⟶𝐵)) | |
12 | 9, 10, 11 | syl2anc 585 | . . . . . . . 8 ⊢ (𝜑 → ((𝐴 × {𝑥}) ∈ (𝐵 ↑m 𝐴) ↔ (𝐴 × {𝑥}):𝐴⟶𝐵)) |
13 | 8, 12 | mpbid 231 | . . . . . . 7 ⊢ (𝜑 → (𝐴 × {𝑥}):𝐴⟶𝐵) |
14 | 13 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐴 × {𝑥}):𝐴⟶𝐵) |
15 | simpr 486 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
16 | 14, 15 | ffvelcdmd 7033 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐴 × {𝑥})‘𝑦) ∈ 𝐵) |
17 | 7, 16 | eqeltrd 2839 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
18 | 17 | ex 414 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
19 | 18 | exlimdv 1937 | . 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 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2942 ∅c0 4281 {csn 4585 × cxp 5630 ⟶wf 6490 ‘cfv 6494 (class class class)co 7352 ↑m cmap 8724 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-fv 6502 df-ov 7355 df-oprab 7356 df-mpo 7357 df-map 8726 |
This theorem is referenced by: mapssbi 43333 |
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