Theorem xpco2 49216

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 6075	. 2 ⊢ Rel ((𝐵 × 𝐶) ∘ 𝐹)
2		relxp 5650	. 2 ⊢ Rel (𝐴 × 𝐶)
3		vex 3446	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
4		vex 3446	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
5	3, 4	breldm 5865	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ dom 𝐹)
6	5	ad2antrl 729	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑥 ∈ dom 𝐹)
7		fdm 6679	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → dom 𝐹 = 𝐴)
9	6, 8	eleqtrd 2839	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑥 ∈ 𝐴)
10		brxp 5681	. . . . . . . . . 10 ⊢ (𝑧(𝐵 × 𝐶)𝑦 ↔ (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
11	10	simprbi 497	. . . . . . . . 9 ⊢ (𝑧(𝐵 × 𝐶)𝑦 → 𝑦 ∈ 𝐶)
12	11	ad2antll 730	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑦 ∈ 𝐶)
13	9, 12	jca 511	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
14	13	ex 412	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ((𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
15	14	exlimdv 1935	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
16	15	imp 406	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
17		ffvelcdm 7035	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
18	17	adantrr 718	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘𝑥) ∈ 𝐵)
19		ffvbr 49215	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥))
20	19	adantrr 718	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → 𝑥𝐹(𝐹‘𝑥))
21		simprr 773	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
22		brxp 5681	. . . . . . 7 ⊢ ((𝐹‘𝑥)(𝐵 × 𝐶)𝑦 ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
23	18, 21, 22	sylanbrc 584	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘𝑥)(𝐵 × 𝐶)𝑦)
24	20, 23	jca 511	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐹(𝐹‘𝑥) ∧ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦))
25		breq2 5104	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑥𝐹𝑧 ↔ 𝑥𝐹(𝐹‘𝑥)))
26		breq1 5103	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧(𝐵 × 𝐶)𝑦 ↔ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦))
27	25, 26	anbi12d 633	. . . . 5 ⊢ (𝑧 = (𝐹‘𝑥) → ((𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) ↔ (𝑥𝐹(𝐹‘𝑥) ∧ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦)))
28	18, 24, 27	spcedv 3554	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦))
29	16, 28	impbida 801	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
30		vex 3446	. . . 4 ⊢ 𝑦 ∈ V
31	3, 30	brco 5827	. . 3 ⊢ (𝑥((𝐵 × 𝐶) ∘ 𝐹)𝑦 ↔ ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦))
32		brxp 5681	. . 3 ⊢ (𝑥(𝐴 × 𝐶)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
33	29, 31, 32	3bitr4g 314	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥((𝐵 × 𝐶) ∘ 𝐹)𝑦 ↔ 𝑥(𝐴 × 𝐶)𝑦))
34	1, 2, 33	eqbrrdiv 5751	1 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 × 𝐶) ∘ 𝐹) = (𝐴 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   class class class wbr 5100   × cxp 5630  dom cdm 5632   ∘ ccom 5636  ⟶wf 6496  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508

This theorem is referenced by:  prcofdiag  49753