Theorem xpco2 49655

Description: Composition of a Cartesian product with a function. (Contributed by Zhi Wang, 25-Nov-2025.)

Assertion

Ref	Expression
xpco2	⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 × 𝐶) ∘ 𝐹) = (𝐴 × 𝐶))

Proof of Theorem xpco2

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 6110	. 2 ⊢ Rel ((𝐵 × 𝐶) ∘ 𝐹)
2		relxp 5679	. 2 ⊢ Rel (𝐴 × 𝐶)
3		vex 3459	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
4		vex 3459	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
5	3, 4	breldm 5898	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ dom 𝐹)
6	5	ad2antrl 740	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑥 ∈ dom 𝐹)
7		fdm 6715	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
8	7	adantr 485	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → dom 𝐹 = 𝐴)
9	6, 8	eleqtrd 2865	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑥 ∈ 𝐴)
10		brxp 5710	. . . . . . . . . 10 ⊢ (𝑧(𝐵 × 𝐶)𝑦 ↔ (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
11	10	simprbi 502	. . . . . . . . 9 ⊢ (𝑧(𝐵 × 𝐶)𝑦 → 𝑦 ∈ 𝐶)
12	11	ad2antll 741	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑦 ∈ 𝐶)
13	9, 12	jca 520	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
14	13	ex 417	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ((𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
15	14	exlimdv 1963	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
16	15	imp 411	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
17		ffvelcdm 7076	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
18	17	adantrr 729	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘𝑥) ∈ 𝐵)
19		ffvbr 49654	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥))
20	19	adantrr 729	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → 𝑥𝐹(𝐹‘𝑥))
21		simprr 784	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
22		brxp 5710	. . . . . . 7 ⊢ ((𝐹‘𝑥)(𝐵 × 𝐶)𝑦 ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
23	18, 21, 22	sylanbrc 594	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘𝑥)(𝐵 × 𝐶)𝑦)
24	20, 23	jca 520	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐹(𝐹‘𝑥) ∧ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦))
25		breq2 5113	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑥𝐹𝑧 ↔ 𝑥𝐹(𝐹‘𝑥)))
26		breq1 5112	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧(𝐵 × 𝐶)𝑦 ↔ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦))
27	25, 26	anbi12d 643	. . . . 5 ⊢ (𝑧 = (𝐹‘𝑥) → ((𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) ↔ (𝑥𝐹(𝐹‘𝑥) ∧ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦)))
28	18, 24, 27	spcedv 3557	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦))
29	16, 28	impbida 812	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
30		vex 3459	. . . 4 ⊢ 𝑦 ∈ V
31	3, 30	brco 5856	. . 3 ⊢ (𝑥((𝐵 × 𝐶) ∘ 𝐹)𝑦 ↔ ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦))
32		brxp 5710	. . 3 ⊢ (𝑥(𝐴 × 𝐶)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
33	29, 31, 32	3bitr4g 317	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥((𝐵 × 𝐶) ∘ 𝐹)𝑦 ↔ 𝑥(𝐴 × 𝐶)𝑦))
34	1, 2, 33	eqbrrdiv 5780	1 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 × 𝐶) ∘ 𝐹) = (𝐴 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143   class class class wbr 5109   × cxp 5659  dom cdm 5661   ∘ ccom 5665  ⟶wf 6532  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544

This theorem is referenced by:  prcofdiag  50192