Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapco2g | Structured version Visualization version GIF version |
Description: Renaming indices in a tuple, with sethood as antecedents. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.) |
Ref | Expression |
---|---|
mapco2g | ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑m 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8427 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐴:𝐶⟶𝐵) | |
2 | fco 6530 | . . . 4 ⊢ ((𝐴:𝐶⟶𝐵 ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵) | |
3 | 1, 2 | sylan 582 | . . 3 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵) |
4 | 3 | 3adant1 1126 | . 2 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵) |
5 | n0i 4298 | . . . . 5 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → ¬ (𝐵 ↑m 𝐶) = ∅) | |
6 | reldmmap 8414 | . . . . . 6 ⊢ Rel dom ↑m | |
7 | 6 | ovprc1 7194 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → (𝐵 ↑m 𝐶) = ∅) |
8 | 5, 7 | nsyl2 143 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐵 ∈ V) |
9 | 8 | 3ad2ant2 1130 | . . 3 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → 𝐵 ∈ V) |
10 | simp1 1132 | . . 3 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → 𝐸 ∈ V) | |
11 | 9, 10 | elmapd 8419 | . 2 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → ((𝐴 ∘ 𝐷) ∈ (𝐵 ↑m 𝐸) ↔ (𝐴 ∘ 𝐷):𝐸⟶𝐵)) |
12 | 4, 11 | mpbird 259 | 1 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑m 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 ∘ ccom 5558 ⟶wf 6350 (class class class)co 7155 ↑m cmap 8405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-map 8407 |
This theorem is referenced by: mapco2 39310 eldioph2 39357 |
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