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 indexes 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 7765 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → 𝐴:𝐶⟶𝐵) | |
2 | fco 5971 | . . . 4 ⊢ ((𝐴:𝐶⟶𝐵 ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵) | |
3 | 1, 2 | sylan 487 | . . 3 ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵) |
4 | 3 | 3adant1 1072 | . 2 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵) |
5 | n0i 3879 | . . . . 5 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → ¬ (𝐵 ↑𝑚 𝐶) = ∅) | |
6 | reldmmap 7753 | . . . . . 6 ⊢ Rel dom ↑𝑚 | |
7 | 6 | ovprc1 6582 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → (𝐵 ↑𝑚 𝐶) = ∅) |
8 | 5, 7 | nsyl2 141 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → 𝐵 ∈ V) |
9 | 8 | 3ad2ant2 1076 | . . 3 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝐷:𝐸⟶𝐶) → 𝐵 ∈ V) |
10 | simp1 1054 | . . 3 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝐷:𝐸⟶𝐶) → 𝐸 ∈ V) | |
11 | 9, 10 | elmapd 7758 | . 2 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝐷:𝐸⟶𝐶) → ((𝐴 ∘ 𝐷) ∈ (𝐵 ↑𝑚 𝐸) ↔ (𝐴 ∘ 𝐷):𝐸⟶𝐵)) |
12 | 4, 11 | mpbird 246 | 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 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ∘ ccom 5042 ⟶wf 5800 (class class class)co 6549 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 |
This theorem is referenced by: mapco2 36296 eldioph2 36343 |
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