mapco2g - Mathbox for Stefan O'Rear

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Theorem mapco2g 42128

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 8861	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐴:𝐶⟶𝐵)
2		fco 6741	. . . 4 ⊢ ((𝐴:𝐶⟶𝐵 ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵)
3	1, 2	sylan 579	. . 3 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵)
4	3	3adant1 1128	. 2 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵)
5		n0i 4329	. . . . 5 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → ¬ (𝐵 ↑_m 𝐶) = ∅)
6		reldmmap 8847	. . . . . 6 ⊢ Rel dom ↑_m
7	6	ovprc1 7453	. . . . 5 ⊢ (¬ 𝐵 ∈ V → (𝐵 ↑_m 𝐶) = ∅)
8	5, 7	nsyl2 141	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐵 ∈ V)
9	8	3ad2ant2 1132	. . 3 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → 𝐵 ∈ V)
10		simp1 1134	. . 3 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → 𝐸 ∈ V)
11	9, 10	elmapd 8852	. 2 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → ((𝐴 ∘ 𝐷) ∈ (𝐵 ↑_m 𝐸) ↔ (𝐴 ∘ 𝐷):𝐸⟶𝐵))
12	4, 11	mpbird 257	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 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ∅c0 4318 ∘ ccom 5676 ⟶wf 6538 (class class class)co 7414 ↑_m cmap 8838

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-map 8840

This theorem is referenced by: mapco2 42129 eldioph2 42176

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