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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 3890 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴) | |
3 | 1, 2 | sylib 207 | . 2 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ 𝐴) |
4 | vex 3176 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | 4 | fvconst2 6374 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ((𝐴 × {𝑥})‘𝑦) = 𝑥) |
6 | 5 | eqcomd 2616 | . . . . . 6 ⊢ (𝑦 ∈ 𝐴 → 𝑥 = ((𝐴 × {𝑥})‘𝑦)) |
7 | 6 | adantl 481 | . . . . 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 7757 | . . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ((𝐴 × {𝑥}) ∈ (𝐵 ↑𝑚 𝐴) ↔ (𝐴 × {𝑥}):𝐴⟶𝐵)) | |
12 | 9, 10, 11 | syl2anc 691 | . . . . . . . 8 ⊢ (𝜑 → ((𝐴 × {𝑥}) ∈ (𝐵 ↑𝑚 𝐴) ↔ (𝐴 × {𝑥}):𝐴⟶𝐵)) |
13 | 8, 12 | mpbid 221 | . . . . . . 7 ⊢ (𝜑 → (𝐴 × {𝑥}):𝐴⟶𝐵) |
14 | 13 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐴 × {𝑥}):𝐴⟶𝐵) |
15 | simpr 476 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
16 | 14, 15 | ffvelrnd 6268 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐴 × {𝑥})‘𝑦) ∈ 𝐵) |
17 | 7, 16 | eqeltrd 2688 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
18 | 17 | ex 449 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
19 | 18 | exlimdv 1848 | . 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 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 {csn 4125 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 |
This theorem is referenced by: mapssbi 38400 |
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