mapcod - Mathbox for Steven Nguyen

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Theorem mapcod 42261

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 8867	. . . 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 8867	. . . 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 8868	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑_m 𝐵) → 𝐹:𝐵⟶𝐴)
10	1, 9	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
11		elmapi 8868	. . . 4 ⊢ (𝐺 ∈ (𝐵 ↑_m 𝐶) → 𝐺:𝐶⟶𝐵)
12	5, 11	syl 17	. . 3 ⊢ (𝜑 → 𝐺:𝐶⟶𝐵)
13	10, 12	fcod 6736	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐶⟶𝐴)
14	4, 8, 13	elmapdd 8860	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑_m 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2109 Vcvv 3464 ∘ ccom 5663 ⟶wf 6532 (class class class)co 7410 ↑_m cmap 8845

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7993 df-2nd 7994 df-map 8847

This theorem is referenced by: evlselv 42577