mapco2g - Mathbox for Stefan O'Rear

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Theorem mapco2g 43302

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.)

**Assertion**
Ref	Expression
mapco2g	⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑_m 𝐸))

**Proof of Theorem mapco2g**
Step	Hyp	Ref	Expression
1		elmapi 8834	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐴:𝐶⟶𝐵)
2		fco 6720	. . . 4 ⊢ ((𝐴:𝐶⟶𝐵 ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵)
3	1, 2	sylan 591	. . 3 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵)
4	3	3adant1 1146	. 2 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵)
5		n0i 4295	. . . . 5 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → ¬ (𝐵 ↑_m 𝐶) = ∅)
6		reldmmap 8820	. . . . . 6 ⊢ Rel dom ↑_m
7	6	ovprc1 7439	. . . . 5 ⊢ (¬ 𝐵 ∈ V → (𝐵 ↑_m 𝐶) = ∅)
8	5, 7	nsyl2 142	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐵 ∈ V)
9	8	3ad2ant2 1150	. . 3 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → 𝐵 ∈ V)
10		simp1 1152	. . 3 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → 𝐸 ∈ V)
11	9, 10	elmapd 8825	. 2 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → ((𝐴 ∘ 𝐷) ∈ (𝐵 ↑_m 𝐸) ↔ (𝐴 ∘ 𝐷):𝐸⟶𝐵))
12	4, 11	mpbird 260	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 1101 = wceq 1563 ∈ wcel 2145 Vcvv 3457 ∅c0 4288 ∘ ccom 5655 ⟶wf 6521 (class class class)co 7400 ↑_m cmap 8812

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-fv 6533 df-ov 7403 df-oprab 7404 df-mpo 7405 df-1st 7974 df-2nd 7975 df-map 8814

This theorem is referenced by: mapco2 43303 eldioph2 43350

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