mapco2g - Mathbox for Stefan O'Rear

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Theorem mapco2g 43068

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 8798	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐴:𝐶⟶𝐵)
2		fco 6694	. . . 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 4294	. . . . 5 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → ¬ (𝐵 ↑_m 𝐶) = ∅)
6		reldmmap 8784	. . . . . 6 ⊢ Rel dom ↑_m
7	6	ovprc1 7407	. . . . 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 8789	. 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 1087 = wceq 1542 ∈ wcel 2114 Vcvv 3442 ∅c0 4287 ∘ ccom 5636 ⟶wf 6496 (class class class)co 7368 ↑_m cmap 8775

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-1st 7943 df-2nd 7944 df-map 8777

This theorem is referenced by: mapco2 43069 eldioph2 43116

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