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Mirrors > Home > ILE Home > Th. List > elmapg | GIF version |
Description: Membership relation for set exponentiation. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
elmapg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapvalg 6652 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ↑𝑚 𝐵) = {𝑔 ∣ 𝑔:𝐵⟶𝐴}) | |
2 | 1 | eleq2d 2247 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶 ∈ {𝑔 ∣ 𝑔:𝐵⟶𝐴})) |
3 | fex2 5380 | . . . . 5 ⊢ ((𝐶:𝐵⟶𝐴 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V) | |
4 | 3 | 3com13 1208 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶:𝐵⟶𝐴) → 𝐶 ∈ V) |
5 | 4 | 3expia 1205 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶:𝐵⟶𝐴 → 𝐶 ∈ V)) |
6 | feq1 5344 | . . . 4 ⊢ (𝑔 = 𝐶 → (𝑔:𝐵⟶𝐴 ↔ 𝐶:𝐵⟶𝐴)) | |
7 | 6 | elab3g 2888 | . . 3 ⊢ ((𝐶:𝐵⟶𝐴 → 𝐶 ∈ V) → (𝐶 ∈ {𝑔 ∣ 𝑔:𝐵⟶𝐴} ↔ 𝐶:𝐵⟶𝐴)) |
8 | 5, 7 | syl 14 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ {𝑔 ∣ 𝑔:𝐵⟶𝐴} ↔ 𝐶:𝐵⟶𝐴)) |
9 | 2, 8 | bitrd 188 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2148 {cab 2163 Vcvv 2737 ⟶wf 5208 (class class class)co 5869 ↑𝑚 cmap 6642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-map 6644 |
This theorem is referenced by: elmapd 6656 mapdm0 6657 elmapi 6664 elmap 6671 map0e 6680 map0g 6682 fdiagfn 6686 ixpssmap2g 6721 map1 6806 mapxpen 6842 infnninf 7116 isomnimap 7129 enomnilem 7130 ismkvmap 7146 enmkvlem 7153 iswomnimap 7158 enwomnilem 7161 hashfacen 10797 omctfn 12424 iscn 13357 iscnp 13359 cndis 13401 ispsmet 13483 ismet 13504 isxmet 13505 elcncf 13720 nnsf 14403 |
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