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 4574 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
3 | 2 | fveq2d 6667 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴)) |
4 | 3 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) = (𝐹‘𝐴)) |
5 | elmapsnd.3 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵) | |
6 | 5 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝐴) ∈ 𝐵) |
7 | 4, 6 | eqeltrd 2910 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) ∈ 𝐵) |
8 | 7 | ralrimiva 3179 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵) |
9 | 1, 8 | jca 512 | . . 3 ⊢ (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)) |
10 | ffnfv 6874 | . . 3 ⊢ (𝐹:{𝐴}⟶𝐵 ↔ (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)) | |
11 | 9, 10 | sylibr 235 | . 2 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵) |
12 | elmapsnd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
13 | snex 5322 | . . . 4 ⊢ {𝐴} ∈ V | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴} ∈ V) |
15 | 12, 14 | elmapd 8409 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m {𝐴}) ↔ 𝐹:{𝐴}⟶𝐵)) |
16 | 11, 15 | mpbird 258 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 {csn 4557 Fn wfn 6343 ⟶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-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: ssmapsn 41355 |
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