mapcod - Mathbox for Steven Nguyen

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Theorem mapcod 43498

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 8899	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑_m 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	3	simpld 500	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
5		mapcod.2	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐵 ↑_m 𝐶))
6		elmapex 8899	. . . 4 ⊢ (𝐺 ∈ (𝐵 ↑_m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
7	5, 6	syl 18	. . 3 ⊢ (𝜑 → (𝐵 ∈ V ∧ 𝐶 ∈ V))
8	7	simprd 501	. 2 ⊢ (𝜑 → 𝐶 ∈ V)
9		elmapi 8900	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑_m 𝐵) → 𝐹:𝐵⟶𝐴)
10	1, 9	syl 18	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
11		elmapi 8900	. . . 4 ⊢ (𝐺 ∈ (𝐵 ↑_m 𝐶) → 𝐺:𝐶⟶𝐵)
12	5, 11	syl 18	. . 3 ⊢ (𝜑 → 𝐺:𝐶⟶𝐵)
13	10, 12	fcod 6760	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐶⟶𝐴)
14	4, 8, 13	elmapdd 8892	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑_m 𝐶))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∈ wcel 2147 Vcvv 3454 ∘ ccom 5660 ⟶wf 6557 (class class class)co 7446 ↑_m cmap 8878

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7449 df-oprab 7450 df-mpo 7451 df-1st 8028 df-2nd 8029 df-map 8880

This theorem is used by: evlselv 43821