Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvmap | Structured version Visualization version GIF version |
Description: Function value for a member of a set exponentiation. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
Ref | Expression |
---|---|
fvmap.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
fvmap.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
fvmap.f | ⊢ (𝜑 → 𝐹 ∈ (𝐴 ↑𝑚 𝐵)) |
fvmap.c | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
Ref | Expression |
---|---|
fvmap | ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝜑 → 𝜑) | |
2 | fvmap.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
3 | fvmap.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐴 ↑𝑚 𝐵)) | |
4 | fvmap.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | fvmap.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
6 | elmapg 7870 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶𝐴)) | |
7 | 4, 5, 6 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶𝐴)) |
8 | 3, 7 | mpbid 222 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴) |
9 | 8 | ffvelrnda 6359 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ 𝐴) |
10 | 1, 2, 9 | syl2anc 693 | 1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 1990 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 |
This theorem is referenced by: ssmapsn 39408 hoidmvle 40814 |
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