mapcod - Mathbox for Steven Nguyen

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Theorem mapcod 42823

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 8825	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑_m 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	3	simpld 498	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
5		mapcod.2	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐵 ↑_m 𝐶))
6		elmapex 8825	. . . 4 ⊢ (𝐺 ∈ (𝐵 ↑_m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → (𝐵 ∈ V ∧ 𝐶 ∈ V))
8	7	simprd 499	. 2 ⊢ (𝜑 → 𝐶 ∈ V)
9		elmapi 8826	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑_m 𝐵) → 𝐹:𝐵⟶𝐴)
10	1, 9	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
11		elmapi 8826	. . . 4 ⊢ (𝐺 ∈ (𝐵 ↑_m 𝐶) → 𝐺:𝐶⟶𝐵)
12	5, 11	syl 17	. . 3 ⊢ (𝜑 → 𝐺:𝐶⟶𝐵)
13	10, 12	fcod 6713	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐶⟶𝐴)
14	4, 8, 13	elmapdd 8818	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑_m 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2141 Vcvv 3453 ∘ ccom 5649 ⟶wf 6513 (class class class)co 7392 ↑_m cmap 8803

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-fv 6525 df-ov 7395 df-oprab 7396 df-mpo 7397 df-1st 7966 df-2nd 7967 df-map 8805

This theorem is referenced by: evlselv 43135