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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 4640 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
3 | 2 | fveq2d 6889 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴)) |
4 | 3 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) = (𝐹‘𝐴)) |
5 | elmapsnd.3 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵) | |
6 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝐴) ∈ 𝐵) |
7 | 4, 6 | eqeltrd 2827 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) ∈ 𝐵) |
8 | 7 | ralrimiva 3140 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵) |
9 | 1, 8 | jca 511 | . . 3 ⊢ (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)) |
10 | ffnfv 7114 | . . 3 ⊢ (𝐹:{𝐴}⟶𝐵 ↔ (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)) | |
11 | 9, 10 | sylibr 233 | . 2 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵) |
12 | elmapsnd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
13 | snex 5424 | . . . 4 ⊢ {𝐴} ∈ V | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴} ∈ V) |
15 | 12, 14 | elmapd 8836 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m {𝐴}) ↔ 𝐹:{𝐴}⟶𝐵)) |
16 | 11, 15 | mpbird 257 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 {csn 4623 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ↑m cmap 8822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-map 8824 |
This theorem is referenced by: ssmapsn 44487 |
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