mapco2g - Mathbox for Stefan O'Rear

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Theorem mapco2g 41323

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 8831	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐴:𝐶⟶𝐵)
2		fco 6731	. . . 4 ⊢ ((𝐴:𝐶⟶𝐵 ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵)
3	1, 2	sylan 581	. . 3 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵)
4	3	3adant1 1131	. 2 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷):𝐸⟶𝐵)
5		n0i 4331	. . . . 5 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → ¬ (𝐵 ↑_m 𝐶) = ∅)
6		reldmmap 8817	. . . . . 6 ⊢ Rel dom ↑_m
7	6	ovprc1 7435	. . . . 5 ⊢ (¬ 𝐵 ∈ V → (𝐵 ↑_m 𝐶) = ∅)
8	5, 7	nsyl2 141	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐵 ∈ V)
9	8	3ad2ant2 1135	. . 3 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → 𝐵 ∈ V)
10		simp1 1137	. . 3 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → 𝐸 ∈ V)
11	9, 10	elmapd 8822	. 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 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4320 ∘ ccom 5676 ⟶wf 6531 (class class class)co 7396 ↑_m cmap 8808

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 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712

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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 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 6487 df-fun 6537 df-fn 6538 df-f 6539 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7962 df-2nd 7963 df-map 8810

This theorem is referenced by: mapco2 41324 eldioph2 41371

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