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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 4415 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
3 | 2 | fveq2d 6450 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴)) |
4 | 3 | adantl 475 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) = (𝐹‘𝐴)) |
5 | elmapsnd.3 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵) | |
6 | 5 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝐴) ∈ 𝐵) |
7 | 4, 6 | eqeltrd 2859 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) ∈ 𝐵) |
8 | 7 | ralrimiva 3148 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵) |
9 | 1, 8 | jca 507 | . . 3 ⊢ (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)) |
10 | ffnfv 6652 | . . 3 ⊢ (𝐹:{𝐴}⟶𝐵 ↔ (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)) | |
11 | 9, 10 | sylibr 226 | . 2 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵) |
12 | elmapsnd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
13 | snex 5140 | . . . 4 ⊢ {𝐴} ∈ V | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴} ∈ V) |
15 | 12, 14 | elmapd 8154 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑𝑚 {𝐴}) ↔ 𝐹:{𝐴}⟶𝐵)) |
16 | 11, 15 | mpbird 249 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑𝑚 {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 Vcvv 3398 {csn 4398 Fn wfn 6130 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ↑𝑚 cmap 8140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-map 8142 |
This theorem is referenced by: ssmapsn 40329 |
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