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Mirrors > Home > MPE Home > Th. List > elmapdd | Structured version Visualization version GIF version |
Description: Deduction associated with elmapd 8830. (Contributed by SN, 29-Jul-2024.) |
Ref | Expression |
---|---|
elmapdd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
elmapdd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
elmapdd.c | ⊢ (𝜑 → 𝐶:𝐵⟶𝐴) |
Ref | Expression |
---|---|
elmapdd | ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑m 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapdd.c | . 2 ⊢ (𝜑 → 𝐶:𝐵⟶𝐴) | |
2 | elmapdd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | elmapdd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | 2, 3 | elmapd 8830 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑m 𝐵) ↔ 𝐶:𝐵⟶𝐴)) |
5 | 1, 4 | mpbird 257 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑m 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ⟶wf 6536 (class class class)co 7404 ↑m cmap 8816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8818 |
This theorem is referenced by: elrspunsn 32505 mapcod 41016 mhmcompl 41070 mhmcoaddmpl 41073 evlsbagval 41088 selvvvval 41107 evlselv 41109 mhphf 41119 |
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