mapcod - Mathbox for Steven Nguyen

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Theorem mapcod 42361

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 8778	. . . 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 8778	. . . 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 8779	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑_m 𝐵) → 𝐹:𝐵⟶𝐴)
10	1, 9	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
11		elmapi 8779	. . . 4 ⊢ (𝐺 ∈ (𝐵 ↑_m 𝐶) → 𝐺:𝐶⟶𝐵)
12	5, 11	syl 17	. . 3 ⊢ (𝜑 → 𝐺:𝐶⟶𝐵)
13	10, 12	fcod 6681	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐶⟶𝐴)
14	4, 8, 13	elmapdd 8771	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑_m 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2113 Vcvv 3437 ∘ ccom 5623 ⟶wf 6482 (class class class)co 7352 ↑_m cmap 8756

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-fv 6494 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7927 df-2nd 7928 df-map 8758

This theorem is referenced by: evlselv 42705