mapcod - Mathbox for Steven Nguyen

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Theorem mapcod 42284

Description: Compose two mappings. (Contributed by SN, 11-Mar-2025.)

**Hypotheses**
Ref	Expression
mapcod.1	⊢ (𝜑 → 𝐹 ∈ (𝐴 ↑_m 𝐵))
mapcod.2	⊢ (𝜑 → 𝐺 ∈ (𝐵 ↑_m 𝐶))

**Assertion**
Ref	Expression
mapcod	⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑_m 𝐶))

**Proof of Theorem mapcod**
Step	Hyp	Ref	Expression
1		mapcod.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐴 ↑_m 𝐵))
2		elmapex 8889	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑_m 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	3	simpld 494	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
5		mapcod.2	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐵 ↑_m 𝐶))
6		elmapex 8889	. . . 4 ⊢ (𝐺 ∈ (𝐵 ↑_m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → (𝐵 ∈ V ∧ 𝐶 ∈ V))
8	7	simprd 495	. 2 ⊢ (𝜑 → 𝐶 ∈ V)
9		elmapi 8890	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑_m 𝐵) → 𝐹:𝐵⟶𝐴)
10	1, 9	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
11		elmapi 8890	. . . 4 ⊢ (𝐺 ∈ (𝐵 ↑_m 𝐶) → 𝐺:𝐶⟶𝐵)
12	5, 11	syl 17	. . 3 ⊢ (𝜑 → 𝐺:𝐶⟶𝐵)
13	10, 12	fcod 6760	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐶⟶𝐴)
14	4, 8, 13	elmapdd 8882	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑_m 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 Vcvv 3479 ∘ ccom 5688 ⟶wf 6556 (class class class)co 7432 ↑_m cmap 8867

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-map 8869

This theorem is referenced by: evlselv 42602