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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elmapsnd | Structured version Visualization version GIF version |
Description: Membership in a set exponentiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
elmapsnd.1 | ⊢ (𝜑 → 𝐹 Fn {𝐴}) |
elmapsnd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
elmapsnd.3 | ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵) |
Ref | Expression |
---|---|
elmapsnd | ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapsnd.1 | . . . 4 ⊢ (𝜑 → 𝐹 Fn {𝐴}) | |
2 | elsni 4542 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
3 | 2 | fveq2d 6649 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴)) |
4 | 3 | adantl 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) = (𝐹‘𝐴)) |
5 | elmapsnd.3 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵) | |
6 | 5 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝐴) ∈ 𝐵) |
7 | 4, 6 | eqeltrd 2890 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) ∈ 𝐵) |
8 | 7 | ralrimiva 3149 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵) |
9 | 1, 8 | jca 515 | . . 3 ⊢ (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)) |
10 | ffnfv 6859 | . . 3 ⊢ (𝐹:{𝐴}⟶𝐵 ↔ (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)) | |
11 | 9, 10 | sylibr 237 | . 2 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵) |
12 | elmapsnd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
13 | snex 5297 | . . . 4 ⊢ {𝐴} ∈ V | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴} ∈ V) |
15 | 12, 14 | elmapd 8403 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m {𝐴}) ↔ 𝐹:{𝐴}⟶𝐵)) |
16 | 11, 15 | mpbird 260 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 {csn 4525 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 |
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 ax-un 7441 |
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-ral 3111 df-rex 3112 df-rab 3115 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-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 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 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 |
This theorem is referenced by: ssmapsn 41845 |
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