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Mirrors > Home > ILE Home > Th. List > elmapd | GIF version |
Description: Deduction form of elmapg 6658. (Contributed by BJ, 11-Apr-2020.) |
Ref | Expression |
---|---|
elmapd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
elmapd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
elmapd | ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapd.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | elmapd.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | elmapg 6658 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴)) | |
4 | 1, 2, 3 | syl2anc 411 | 1 ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∈ wcel 2148 ⟶wf 5211 (class class class)co 5872 ↑𝑚 cmap 6645 |
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 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-fv 5223 df-ov 5875 df-oprab 5876 df-mpo 5877 df-map 6647 |
This theorem is referenced by: elmapssres 6670 mapss 6688 mapen 6843 mapxpen 6845 fodjuf 7140 ismkvnex 7150 ismhm 12785 cnpdis 13613 bj-charfunbi 14423 nninfself 14622 isomninnlem 14638 trilpolemlt1 14649 iswomninnlem 14657 iswomni0 14659 ismkvnnlem 14660 |
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