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 4586 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
3 | 2 | fveq2d 6676 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴)) |
4 | 3 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) = (𝐹‘𝐴)) |
5 | elmapsnd.3 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵) | |
6 | 5 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝐴) ∈ 𝐵) |
7 | 4, 6 | eqeltrd 2915 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) ∈ 𝐵) |
8 | 7 | ralrimiva 3184 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵) |
9 | 1, 8 | jca 514 | . . 3 ⊢ (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)) |
10 | ffnfv 6884 | . . 3 ⊢ (𝐹:{𝐴}⟶𝐵 ↔ (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)) | |
11 | 9, 10 | sylibr 236 | . 2 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵) |
12 | elmapsnd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
13 | snex 5334 | . . . 4 ⊢ {𝐴} ∈ V | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴} ∈ V) |
15 | 12, 14 | elmapd 8422 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m {𝐴}) ↔ 𝐹:{𝐴}⟶𝐵)) |
16 | 11, 15 | mpbird 259 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 {csn 4569 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 |
This theorem is referenced by: ssmapsn 41486 |
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